tag:blogger.com,1999:blog-59962684921563642012024-03-15T07:02:20.063-07:00MuonRayScience, Technology, Investigation, Experimentation and Visualisation: Irradiate Yourself with MuonRay!MuonRayhttp://www.blogger.com/profile/03712859045968965104noreply@blogger.comBlogger84125tag:blogger.com,1999:blog-5996268492156364201.post-9196388624492054802024-02-27T06:00:00.000-08:002024-02-27T06:00:05.760-08:00Thoughts on a Resilient Network Model of Learning Delivery and General Education<p><b>Introduction</b></p><p>In this writing I wish to outline a teaching philosophy based on a model of teaching with the
concept of resilience in mind. What this means is a forging a framework as a dynamic and
adaptive system of learning based on generating learning materials from a subject curriculum,
reflecting on the work resources created and forming active links of information exchange
between the reservoir of material and the learning goals of an individual student and student
community, links that have the capacity to be changed, updated and/or given new avenues to
further learning all the while being adaptive enough so that the student community is bolstered
and made resilient themselves to make behavioral changes in response to outside shocks. This, I
am convinced, is a reasonable way to ensure that local and global changes to learning routines
are not disastrous to the overall protocol of education. </p><p><b>Resilient Model of Learning Delivery </b></p><p>Teaching can be defined as an action that conveys a meaningful exchange of information,
using work done by the teacher and the student community, to give an impact that establishes
links to educational material and students' ability to recall and use that material in a dynamic and
coherent way to solve problems in a complex environment. The students must be linked with
material, through education, on the stage of the complex environment itself and so the process
must be a resilient one to external factors.
Learning then is also about strengthening the links made in a given educational setting, i.e. a
class, and making it transferable to another setting, such as a project, exam or work setting. The
links we may want to make resilient are as varied as the curriculum itself, not limited to the
ability to remember how to problem solve but to deconstruct information, construct models,
share information or integrate new information. </p><p>These links are made resilient within the
educational process as they are practiced, refined, subject to shifts in priorities based on local and
global factors or in fact broken and replaced/restored as need arises.
A resilient educational process therefore must have structure, with a plastic and versatile
memory, where the conventional focal point of education is established by the nature of the
external complex society. This varies quite a lot given the nature of different societies across the
world and across time, goal orientation and/or grade validation may not be the focus or purpose
of education in one society but may have a local basis for existence in another. This is a way of
saying goals can vary in the structure of society and a student's own goals, for example learning
a trade or skill by “learning by doing” rather than pure theory. </p><p>Any educational protocol that can call itself resilient must have its actions linked to outcomes
that can take a form of measurable evidence to provide coherent feedback of the learning
resources over the course of time. From raw evidence such as grading outcomes, sampling of
students focus of learning materials (i.e notes, requests or download of study content over time
etc.) as well as feedback a specific framework of reflection of learning outcomes, management of
content, judgement and choices based on the delivery of the subject matter in class, examinations
and projects can be created. This can form an adaptive memory of teaching a subject as the
subject itself changes or as the student community changes or other unforeseen changes occur.
In a resilient model, modifications to education delivery may be changed based on the measured
changes, with certain links reflecting the action of delivery from a teaching resource to a student
audience being broken, replace or switched from one strategy to another over time. </p><p><b>Model Diagram</b></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEghu4dxsXn5ynrj31rMS9HImUqp3aTXuw7bwqdnY0yL6prmPtsRsWwGjhw5Nd7nZ9Su8qWU6TmhYlbqf3z5HIVPqkckMJkJW1epj8qDglBIAGUsG_s8YeuLemslze80qpV0xmh-D8cSVJjgsGdXc2Dmyy9fLnXzkphCAKW_NVWP5l6Rm7ylJmyVOtwvSo3Z/s1262/curriculum%20model.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1262" data-original-width="1102" height="628" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEghu4dxsXn5ynrj31rMS9HImUqp3aTXuw7bwqdnY0yL6prmPtsRsWwGjhw5Nd7nZ9Su8qWU6TmhYlbqf3z5HIVPqkckMJkJW1epj8qDglBIAGUsG_s8YeuLemslze80qpV0xmh-D8cSVJjgsGdXc2Dmyy9fLnXzkphCAKW_NVWP5l6Rm7ylJmyVOtwvSo3Z/w548-h628/curriculum%20model.png" width="548" /></a></div><p>In this model, perhaps changes, or indeed shocks, to any learning delivery system are based on local factors in the class setting, greater university setting or global factors due to
unprecedented events. </p><p></p><ul style="text-align: left;"><li>In local cases we could have 2 scenarios: </li></ul><p></p><p><br /></p><p>First, we could consider an individual student or student community engaged in learning
behavior wherein they form a certain set of links to and from the learning resource material and
the outcomes in a curriculum. Students as individual may see the curriculum as a well of sorts
that has to be filled via resilient links that they would create to reach a certain level of mastery. </p><p>Overall, a student community would be given an expectation to reach a certain level by the end
of a class or course, for example as listed in a class/course syllabus. The links established in the
curriculum may be treated as equal in an unchanging background but in an instance of an exam
being scheduled or a group project proposed the student community and Indvidual's will form a
shift in behavior in what links in the curriculum well will be favored and thus the links
themselves must be made resilient enough to bolster this change in priorities. </p><p>In a second example, a module may have an element that was in one year confined only to theory
but based on a reflection generated by evidence gathered of learning outcomes in tandem with
new learning resources being made available, old links from learning resources to class delivery
are switched from being pure theory to theory and practical. This can cause changes that may, for
example, go on to replace the theory component altogether in a fashion which may not have been
considered without updating from a reflection. </p><p><br /></p><p></p><ul style="text-align: left;"><li>In global cases one can think of a very general set of scenarios: </li></ul><p></p><p>These scenarios would be global in scope, caused by changes in society, technology economics,
job markets and unprecedented occurrences due to changing demographics, natural and human induced disasters, diminishing returns on growth and increased societal pressures. A model based
on resilience is not an abstraction in these cases and in several instances one can think of
education resources for instance having to be made streamlined, efficient but highly resilient.
For example, during a natural disaster a course that was in the practical and hands-on domain
may be moved entirely online and remote. Lets say students in first year learning practical
electronic circuit building. </p><p>Such a course may have had online simulation and theory resources
were made previously available and would ideally have been made resilient enough to
effectively be “switched on” and have the link between the resource and student acquisition strengthened via more active reflection so that the link would be resilient and be an avenue for
learning goals that yield outcomes to feedback into further reflection and refinement over time. </p><p><br /></p><p><b>Final Statement </b></p><p>The model I have described, and illustrated as a diagram below, is in many cases, fairly
generative and is one that itself is a product of my own personal reflection and rumination about
how to teach varied subject matter, mostly in the field of science, engineering and mathematics,
in a highly complex and dynamic emerging local and global community. It is my sincere
worldview however that resilient networks within institutional systems allow for resilient
communities to be able to emerge and withstand local and global shifts behavior that is
inherently risky but made safer to implement if the links between informational resources and
active teaching is continually updated in an adaptive way.</p>MuonRayhttp://www.blogger.com/profile/03712859045968965104noreply@blogger.com0tag:blogger.com,1999:blog-5996268492156364201.post-85622256158222459622023-12-11T12:48:00.000-08:002023-12-11T13:04:44.901-08:00Building Bridges between Synchronization and Quantum Entanglement in Networks<iframe frameborder="0" height="270" src="https://youtube.com/embed/E6kKQZpWRHU?si=Y4VVaV5DQqQjdF-g" width="480"></iframe><div><br /></div><div><br /></div><div><span style="background-color: rgba(0, 0, 0, 0.05); color: #0f0f0f; font-family: Roboto, Arial, sans-serif; font-size: 14px; white-space-collapse: preserve;">The similarities between so-called classical synchronization and quantum entanglement is not lost on some of the emergent, self-similar behavior, that exists in these systems. This can be seen in the systems themselves, separately. Perturbations in quantum systems lead to discontinuities that can lead to decoherence however systems of entangled quantum oscillators can also display error-correction in certain topological states, such as toric and/or surface codes. Classical systems meanwhile display so-called chimera states that exist as a "phase state" between order and disorder to an extent that these states can actually steer a self-organised system back from a chaotic edge and maintain itself durable. I have explored these concepts in a previous video:
However here I wish to showcase some of the actual simulations I've done with software and hardware which does not necessarily take us into using the exotic systems found in a quantum optics lab.
Metaheuristic algorithms meanwhile that port and parse some of the measurement spaces found in quantum systems, i.e. the Poincare/Bloch/Riemann Sphere and represent them as "squashed" pseudo-quantum states represented as HSV values say can nevertheless display some of the "quantumness" which can be described using, among other things, the path integral formalism of quantum mechanics and even resembles the behavior of real-world quantum states of matter such as entangled networks, Bose-Einstein condensates, currents and flows of Cooper pairs in superconductors etc.
From all of this we could very well as, are all of these variations on a common physical theme? Another question we could ask in this research is, at what scale does entanglement end and synchronization begin and vice versa?</span></div>MuonRayhttp://www.blogger.com/profile/03712859045968965104noreply@blogger.com0tag:blogger.com,1999:blog-5996268492156364201.post-31869752918453174552023-11-26T11:03:00.000-08:002023-11-26T11:32:09.838-08:00Experiment in making Periodically-Poled KTP (ppKTP) for Quantum Optics Research using off-the-shelf KTP<iframe frameborder="0" height="270" src="https://youtube.com/embed/EIHTiRc_rlg?si=deoR90FwM80l_i-X" style="background-image: url(https://i.ytimg.com/vi/EIHTiRc_rlg/hqdefault.jpg);" width="480"></iframe><div><br /></div><div><br /></div><div><span face="Roboto, Arial, sans-serif" style="background-color: rgba(255, 255, 255, 0.1); color: #2b00fe; font-size: 14px; white-space-collapse: preserve;"><b>a big focus of my research is buying bespoke optics components and repurposing them for quantum optics experiments and the relative abundance of KTP (potassium titanyl phosphate) along with micro electrodes motivated me (among others things) to try and see if current setups using BBO (beta-barium borate) can be enhanced with modified components that would otherwise be prohibitively expensive to integrate into existing systems.
Measuring the degree of polarization-entanglement is the next step and it too uses a lot of off-the-shelf technology but, again, restructured for a different purpose. In many instances the research goal is to create new architectures in networks that already exist in the classical domain. As a point of fact a lot of my research has taken me into bridging classical networks and their components into the emerging field of "quantum" networks. In many ways the components for "quantum" networks already exist, the networks themselves just have to be tailored with a different approach than before.</b></span></div>MuonRayhttp://www.blogger.com/profile/03712859045968965104noreply@blogger.com0tag:blogger.com,1999:blog-5996268492156364201.post-26064991867817286332022-10-20T11:49:00.005-07:002022-10-20T11:49:56.551-07:00Capturing Emission Spectra from a Lightning Bolt during a Thunderstorm i...<iframe frameborder="0" height="270" src="https://youtube.com/embed/GBzAdv3MvC0" width="480"></iframe><div><br /></div><div><br /></div><div><span style="background-color: white; color: #0d0d0d; font-family: Roboto, Noto, sans-serif; font-size: 15px; white-space: pre-wrap;">I've always wanted to see if I could capture at least a low resolution image of emission spectra from a clear shot of a lightning bolt using a diffraction grating and a camera and a thunderstorm last night presented a rare opportunity to test this idea!</span></div><div><span style="background-color: white; color: #0d0d0d; font-family: Roboto, Noto, sans-serif; font-size: 15px; white-space: pre-wrap;"><br /></span></div><div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj-u2s3GFFGNpjaBrJqVdhiCHEd2GLCpId7bYZKqUJmkYIbI22RSJK9nERvauCez1DtkgvUb3wRWAXtT8cwD2b93BAMmDvQtzJSsOixO_X8X1-7gQXdkPB3_5HGBIf3rwcX-oNj7o-EJKt9Q6syZ2BzHWiNLpCU2eoeW4FmgorOYj7EWOPal-NSNz8D0A/s1234/imageedit_4_3651423438.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="702" data-original-width="1234" height="228" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj-u2s3GFFGNpjaBrJqVdhiCHEd2GLCpId7bYZKqUJmkYIbI22RSJK9nERvauCez1DtkgvUb3wRWAXtT8cwD2b93BAMmDvQtzJSsOixO_X8X1-7gQXdkPB3_5HGBIf3rwcX-oNj7o-EJKt9Q6syZ2BzHWiNLpCU2eoeW4FmgorOYj7EWOPal-NSNz8D0A/w400-h228/imageedit_4_3651423438.png" width="400" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhjE6hDqCCe6OmAblM6LK_DwBCgp-AZsu-DEk_bW7i-F0h0-bnOvIRd9j_gRqnqS1BXUPN9nbUNSOJZEakIZKnwtojmMl79fkLGQIjs_41X_B9h8XFLRzuc-WW_CGX010Dq9S58dUrNaY7F9vAoaVPbcc_E8eJ7dO5Fnu8-MxmD-HTrv4qWC_JCRtLjRA/s546/thanks%20for%20watching.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="283" data-original-width="546" height="332" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhjE6hDqCCe6OmAblM6LK_DwBCgp-AZsu-DEk_bW7i-F0h0-bnOvIRd9j_gRqnqS1BXUPN9nbUNSOJZEakIZKnwtojmMl79fkLGQIjs_41X_B9h8XFLRzuc-WW_CGX010Dq9S58dUrNaY7F9vAoaVPbcc_E8eJ7dO5Fnu8-MxmD-HTrv4qWC_JCRtLjRA/w640-h332/thanks%20for%20watching.png" width="640" /></a></div><br /><span style="background-color: white; color: #0d0d0d; font-family: Roboto, Noto, sans-serif; font-size: 15px; white-space: pre-wrap;"><br /></span></div>MuonRayhttp://www.blogger.com/profile/03712859045968965104noreply@blogger.com0tag:blogger.com,1999:blog-5996268492156364201.post-89184889149476396632022-04-04T14:34:00.001-07:002022-04-04T14:34:09.419-07:00Drone Environmental Monitoring of The Bandama Caldera Thermophilic Forest - Using Python for NDVI, ENDVI, SAVI Image Processing<iframe frameborder="0" height="270" src="https://youtube.com/embed/avDXTtGd1Zw" width="480"></iframe><div><span style="background-color: white; color: #0d0d0d; font-family: Roboto, Noto, sans-serif; font-size: 15px; white-space: pre-wrap;">In a recent trip to Gran Canaria, I visited one of my favorite natural landscapes there; the Bandama Caldera which hosts a diverse variety of plant and animal life. Since the weather was clear and the illumination was good I decided to deploy my drone to perform some visual inspection and near-infrared environmental monitoring with NDVI, ENDVI and SAVI metrics extracted using my github codes:https://github.com/MuonRay/PythonNDVI</span></div><div><span style="background-color: white; color: #0d0d0d; font-family: Roboto, Noto, sans-serif; font-size: 15px; white-space: pre-wrap;"><br /></span></div><div><span style="background-color: white; color: #0d0d0d; font-family: Roboto, Noto, sans-serif; font-size: 15px; white-space: pre-wrap;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEioPo3bFgGsAcSQUp5kic9h4LF0bGGlG0Lpo0GmZrHBLh1JSzk1jlwFhPgb4QZzJ_whzEohCI20XPDrZW_vIsVtewBk5rL7NMP0TZbnqJ6EUUthVw-CgTYUUqv6aYj4H_BUIPQhUUgs8_3PEcxFtgy5enKT84Rl4CrXhlOp0x3UmOjsOegB8sQ2o8MTXw/s2789/InfraBlueNDVI876.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1811" data-original-width="2789" height="208" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEioPo3bFgGsAcSQUp5kic9h4LF0bGGlG0Lpo0GmZrHBLh1JSzk1jlwFhPgb4QZzJ_whzEohCI20XPDrZW_vIsVtewBk5rL7NMP0TZbnqJ6EUUthVw-CgTYUUqv6aYj4H_BUIPQhUUgs8_3PEcxFtgy5enKT84Rl4CrXhlOp0x3UmOjsOegB8sQ2o8MTXw/s320/InfraBlueNDVI876.jpg" width="320" /></a></div><br /></span></div><div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEirwuEVEoaBpa4ILGfL7VIyx0hVz8dZO8voOGkM_WAkfSpLY68LRKIWSKW6IF1X0gn1s8Cdy8Ayhu3je0KemaIYePfVtvy5EVIdZScbvb6ycnbiEq4BM0C7yxuJ2FuIGqejktNwiVcERQppo7b00e5zcj2cLLo7tzCwur_l1A25ksNN0t_EvRTiB_kcHA/s2789/DJI_0876.jpeg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1811" data-original-width="2789" height="208" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEirwuEVEoaBpa4ILGfL7VIyx0hVz8dZO8voOGkM_WAkfSpLY68LRKIWSKW6IF1X0gn1s8Cdy8Ayhu3je0KemaIYePfVtvy5EVIdZScbvb6ycnbiEq4BM0C7yxuJ2FuIGqejktNwiVcERQppo7b00e5zcj2cLLo7tzCwur_l1A25ksNN0t_EvRTiB_kcHA/s320/DJI_0876.jpeg" width="320" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjyJZy1hTONfnQ9C3-Js3RiyWxbOTO--1YvO8bkzDM8gGR1hDGc5Q8TQlZO-iHaCZhmOqoLVAb0t_OkUMdB89eRkI_e62Ed-hd5BHUYkF0JXHsxPLk3SjsmrxxKZaaS771S50AITBF0xwe1zVfjy0FeBmnbFQD5MT22TnbeQvTsZSQSZd9JTIdxMu-fVQ/s2789/DJI_0876.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1811" data-original-width="2789" height="208" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjyJZy1hTONfnQ9C3-Js3RiyWxbOTO--1YvO8bkzDM8gGR1hDGc5Q8TQlZO-iHaCZhmOqoLVAb0t_OkUMdB89eRkI_e62Ed-hd5BHUYkF0JXHsxPLk3SjsmrxxKZaaS771S50AITBF0xwe1zVfjy0FeBmnbFQD5MT22TnbeQvTsZSQSZd9JTIdxMu-fVQ/s320/DJI_0876.JPG" width="320" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><br /></div><span style="background-color: white; color: #0d0d0d; font-family: Roboto, Noto, sans-serif; font-size: 15px; white-space: pre-wrap;"><div><span style="background-color: white; color: #0d0d0d; font-family: Roboto, Noto, sans-serif; font-size: 15px; white-space: pre-wrap;"><br /></span></div>Using my freely available (<b>free for non-profit use</b>) Python codes I can convert near-infrared images captured using my modified NIR camera with my custom made filters to perform remote sensing using my drone. I am continually working on refining the method and analysis of performing NDVI in the field and will soon be able to perform the analysis in real time using my phone which can run my python scripts using a mobile app. I hope to showcase this soon in a future video.</span></div>MuonRayhttp://www.blogger.com/profile/03712859045968965104noreply@blogger.com0tag:blogger.com,1999:blog-5996268492156364201.post-1405762183795286642022-02-03T17:24:00.000-08:002022-02-03T17:24:05.060-08:00Quantum Encryption of Images in Python using Bitwise XOR and a QRNG<h1 style="text-align: left;"><u><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEgMgEchF0cZXH0LfvQ9aZWnWFShc7vba1nHzlmHiRhn3v4f4i2UQal3XDbjkasp886vW-QNyE2YN4bdJmam59Ey3gZfvRvGto1yjC_jiBqSozWkFwidy8xqu-KUSuTKRuaEErnp50QE9oD4N8CAAgYAUdDr2sYrShO6gfJ0R1_NSeFYyM5tjIDWHsRzjg=s1444" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="958" data-original-width="1444" height="331" src="https://blogger.googleusercontent.com/img/a/AVvXsEgMgEchF0cZXH0LfvQ9aZWnWFShc7vba1nHzlmHiRhn3v4f4i2UQal3XDbjkasp886vW-QNyE2YN4bdJmam59Ey3gZfvRvGto1yjC_jiBqSozWkFwidy8xqu-KUSuTKRuaEErnp50QE9oD4N8CAAgYAUdDr2sYrShO6gfJ0R1_NSeFYyM5tjIDWHsRzjg=w500-h331" width="500" /></a></div><br /><span style="font-size: large;"><br /></span></u></h1><h1 style="text-align: left;"><u><span style="font-size: large;">XOR Cipher in Standard Cryptography</span></u></h1><p><span style="font-family: arial;">In Cryptography, the exclusive OR or XOR Cipher is an additive method of encryption of a string of data using a particular key.</span></p><p><span style="font-family: arial;"><span style="background-color: white; color: #202122;">The XOR operator is extremel</span><span style="background-color: white; color: #202122;">y common as a component in more complex ciphers. By itself, using a constant repeating key, a simple XOR cipher can trivially be broken using frequency analysis for pattern recognition as a means to reconstr</span><span style="background-color: white; color: #202122;">uct the key generation process, if not made using a Pseudo-Random Number Generator for instance. If the content of any message can be guessed or otherwise known then the key can be revealed.</span></span></p><p style="background-color: white; color: #202122; margin: 0.5em 0px;"><span style="font-family: arial;">The primary merit of the XOR Cipher is that it is simple to implement, with the XOR operation being computationally inexpensive. </span></p><p><span style="font-family: arial;">The XOR Cipher itself can be implemented using a XOR Logic Gate in a Bitwise Function.</span></p><p style="background-color: white; color: #202122; margin: 0.5em 0px;"><span style="font-family: arial;">an encryption algorithm that uses the XOR operates according to the principles:</span></p><dl style="background-color: white; color: #202122; font-family: sans-serif; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span style="font-size: medium;">A <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px, 1px, 1px, 1px); display: none; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math alttext="{\displaystyle \oplus }" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \oplus }</annotation></semantics></math></span><img alt="\oplus" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b16e2bdaefee9eed86d866e6eba3ac47c710f60" style="border: 0px; display: inline-block; height: 2.176ex; vertical-align: -0.505ex; width: 1.808ex;" /></span> 0 = A,</span></dd><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span style="font-size: medium;">A <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px, 1px, 1px, 1px); display: none; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math alttext="{\displaystyle \oplus }" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \oplus }</annotation></semantics></math></span><img alt="\oplus" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b16e2bdaefee9eed86d866e6eba3ac47c710f60" style="border: 0px; display: inline-block; height: 2.176ex; vertical-align: -0.505ex; width: 1.808ex;" /></span> A = 0,</span></dd><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span style="font-size: medium;">A <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px, 1px, 1px, 1px); display: none; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math alttext="{\displaystyle \oplus }" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \oplus }</annotation></semantics></math></span><img alt="\oplus" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b16e2bdaefee9eed86d866e6eba3ac47c710f60" style="border: 0px; display: inline-block; height: 2.176ex; vertical-align: -0.505ex; width: 1.808ex;" /></span> B = B <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px, 1px, 1px, 1px); display: none; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math alttext="{\displaystyle \oplus }" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \oplus }</annotation></semantics></math></span><img alt="\oplus" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b16e2bdaefee9eed86d866e6eba3ac47c710f60" style="border: 0px; display: inline-block; height: 2.176ex; vertical-align: -0.505ex; width: 1.808ex;" /></span> A,</span></dd><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span style="font-size: medium;">(A <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px, 1px, 1px, 1px); display: none; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math alttext="{\displaystyle \oplus }" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \oplus }</annotation></semantics></math></span><img alt="\oplus" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b16e2bdaefee9eed86d866e6eba3ac47c710f60" style="border: 0px; display: inline-block; height: 2.176ex; vertical-align: -0.505ex; width: 1.808ex;" /></span> B) <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px, 1px, 1px, 1px); display: none; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math alttext="{\displaystyle \oplus }" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \oplus }</annotation></semantics></math></span><img alt="\oplus" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b16e2bdaefee9eed86d866e6eba3ac47c710f60" style="border: 0px; display: inline-block; height: 2.176ex; vertical-align: -0.505ex; width: 1.808ex;" /></span> C = A <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px, 1px, 1px, 1px); display: none; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math alttext="{\displaystyle \oplus }" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \oplus }</annotation></semantics></math></span><img alt="\oplus" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b16e2bdaefee9eed86d866e6eba3ac47c710f60" style="border: 0px; display: inline-block; height: 2.176ex; vertical-align: -0.505ex; width: 1.808ex;" /></span> (B <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px, 1px, 1px, 1px); display: none; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math alttext="{\displaystyle \oplus }" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \oplus }</annotation></semantics></math></span><img alt="\oplus" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b16e2bdaefee9eed86d866e6eba3ac47c710f60" style="border: 0px; display: inline-block; height: 2.176ex; vertical-align: -0.505ex; width: 1.808ex;" /></span> C),</span></dd><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span style="font-size: medium;">(B <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px, 1px, 1px, 1px); display: none; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math alttext="{\displaystyle \oplus }" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \oplus }</annotation></semantics></math></span><img alt="\oplus" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b16e2bdaefee9eed86d866e6eba3ac47c710f60" style="border: 0px; display: inline-block; height: 2.176ex; vertical-align: -0.505ex; width: 1.808ex;" /></span> A) <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px, 1px, 1px, 1px); display: none; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math alttext="{\displaystyle \oplus }" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \oplus }</annotation></semantics></math></span><img alt="\oplus" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b16e2bdaefee9eed86d866e6eba3ac47c710f60" style="border: 0px; display: inline-block; height: 2.176ex; vertical-align: -0.505ex; width: 1.808ex;" /></span> A = B <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px, 1px, 1px, 1px); display: none; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math alttext="{\displaystyle \oplus }" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \oplus }</annotation></semantics></math></span><img alt="\oplus" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b16e2bdaefee9eed86d866e6eba3ac47c710f60" style="border: 0px; display: inline-block; height: 2.176ex; vertical-align: -0.505ex; width: 1.808ex;" /></span> 0 = B,</span></dd></dl><p style="background-color: white; color: #202122; margin: 0.5em 0px;"><span style="font-family: arial;">where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px, 1px, 1px, 1px); display: none; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math alttext="{\displaystyle \oplus }" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle \oplus }</annotation></semantics></math></span><img alt="\oplus" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b16e2bdaefee9eed86d866e6eba3ac47c710f60" style="border: 0px; display: inline-block; height: 2.176ex; margin: 0px; vertical-align: -0.505ex; width: 1.808ex;" /></span> denotes the exclusive OR (XOR) logic operation.<span style="white-space: nowrap;"> </span>This operation is sometimes called modulus 2 addition (or subtraction, which is identical).<span style="white-space: nowrap;"> </span>With this logic, a string of text can be encrypted by applying the bitwise XOR operator to every character using a given key. </span></p><p style="background-color: white; color: #202122; margin: 0.5em 0px;"><span style="font-family: arial;">To decrypt the output, merely reapplying the XOR function with the key will remove the cipher, as the XOR operation is its own inverse.</span></p><p style="background-color: white; color: #202122; margin: 0.5em 0px;"><span style="font-family: arial;"><br /></span></p><p style="background-color: white; color: #202122; margin: 0.5em 0px;"><span style="font-family: arial;">In any of these ciphers, the XOR operator is vulnerable to a known-plaintext attack, since </span></p><p style="background-color: white; color: #202122; font-family: sans-serif; margin: 0.5em 0px;"><span style="font-size: medium;"><i>plaintext</i> <span class="mwe-math-element"><img alt="\oplus" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b16e2bdaefee9eed86d866e6eba3ac47c710f60" style="border: 0px; display: inline-block; height: 2.176ex; margin: 0px; vertical-align: -0.505ex; width: 1.808ex;" /></span> <i>ciphertext</i> = <i>key</i>.</span></p><p style="background-color: white; color: #202122; margin: 0.5em 0px;"><span style="font-family: arial;"><br /></span></p><p style="background-color: white; margin: 0.5em 0px;"><span style="color: #202122; font-family: arial;">If the key used is random, then the XOR operation will in a sense preserve the randomness of the key in the encrypted data. The result is a random stream cipher. If a key is generated using a truly random number source, such as a quantum random number generator, the result is the generation of one-time pad using a single use, truly unique key that is in principle impossible to crack. This is the motivation for generating QRNGs which can combat the emergence of quantum computation that could, in principle, perform known-plaintext attacks of stream cipher encryption in polynomial time. Quantum Encryption is the only known way to combat this threat.</span></p><p style="background-color: white; margin: 0.5em 0px;">We can examine 2 methods to create encryption keys: using a Quantum Random Number Generator and by Using quantum correlated images captured using entangled photons.</p><p style="background-color: white; margin: 0.5em 0px;"><br /></p><p style="background-color: white; margin: 0.5em 0px;"><br /></p><p style="background-color: white; color: #202122; font-family: sans-serif; font-size: 14px; margin: 0.5em 0px;"><br /></p><h1 style="background-color: white; color: #202122; font-family: sans-serif; margin: 0.5em 0px; text-align: left;"><u><span style="font-size: large;">Quantum Random Number Generation</span></u></h1><div><p style="background-color: white; border: 0px; box-sizing: inherit; clear: both; color: #232629; font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", "Liberation Sans", sans-serif; font-size: 15px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin-bottom: var(--s-prose-spacing); margin-left: 0px; margin-right: 0px; margin-top: 0px; padding: 0px; vertical-align: baseline;">As previously discussed, the random stream XOR cipher can be generated using a random number generator. pseudo-random number generators (PRNGs) are build on algorithms involving some kind of recursive method starting from a base value that is determined by an input called the "seed". The default PRNG in most statistical software (R, Python, Stata, etc.) is the Mersenne Twister algorithm MT19937, which is set out in Matsumoto and Nishimura (1998). This is a complicated algorithm, so it would be best to read the paper on it if you want to know how it works in detail. In this particular algorithm, there is a recurrence relation of degree <span class="MathJax_Preview" color="inherit" style="background: inherit; border: 0px; box-sizing: inherit; font-family: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;"></span><span class="MathJax" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>" id="MathJax-Element-12-Frame" role="presentation" style="border: 0px; box-sizing: inherit; direction: ltr; display: inline; float: none; font-family: inherit; font-stretch: inherit; font-variant: inherit; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; position: relative; vertical-align: baseline; white-space: nowrap; word-spacing: normal;" tabindex="0"><nobr aria-hidden="true" style="border: 0px; box-sizing: inherit; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-37" style="border: 0px; box-sizing: content-box; display: inline-block; font-family: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.693em;"><span style="border: 0px; box-sizing: content-box; display: inline-block; font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.578em;"><span style="border: 0px; box-sizing: content-box; clip: rect(1.67em, 1000.58em, 2.474em, -999.997em); font-family: inherit; font-size: 17.4px; font-stretch: inherit; 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font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; height: 2.302em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; box-sizing: content-box; display: inline-block; font-family: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; height: 0.67em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.063em; width: 0px;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation" style="border: 0px; box-sizing: content-box; clip: rect(1px, 1px, 1px, 1px); display: inline; font-family: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; height: 1px; left: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; top: 0px; transition: none 0s ease 0s; user-select: none; vertical-align: 0px; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span>, and your input seed is an initial set of vectors <span class="MathJax_Preview" color="inherit" style="background: inherit; border: 0px; box-sizing: inherit; font-family: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;"></span><span class="MathJax" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">x</mi></mrow><mn>0</mn></msub><mo>,</mo><msub><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">x</mi></mrow><mn>1</mn></msub><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><msub><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">x</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>&#x2212;</mo><mn>1</mn></mrow></msub></math>" id="MathJax-Element-13-Frame" role="presentation" style="border: 0px; box-sizing: inherit; direction: ltr; display: inline; float: none; font-family: inherit; font-stretch: inherit; font-variant: inherit; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; position: relative; vertical-align: baseline; white-space: nowrap; word-spacing: normal;" tabindex="0"><nobr aria-hidden="true" style="border: 0px; box-sizing: inherit; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-40" style="border: 0px; box-sizing: content-box; display: inline-block; font-family: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 7.876em;"><span style="border: 0px; box-sizing: content-box; display: inline-block; font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 6.784em;"><span style="border: 0px; box-sizing: content-box; clip: rect(1.727em, 1006.78em, 2.761em, -999.997em); font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.353em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-41" style="border: 0px; box-sizing: content-box; display: inline; font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; 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padding: 0px; position: absolute; top: -4.02em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-43" style="border: 0px; box-sizing: content-box; display: inline; font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-44" style="border: 0px; box-sizing: content-box; display: inline; font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-45" style="border: 0px; box-sizing: content-box; display: inline; font-family: MathJax_Main-bold; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">x</span></span></span><span style="border: 0px; box-sizing: content-box; display: inline-block; font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; height: 4.026em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: content-box; font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; left: 0.635em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.848em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-46" style="border: 0px; box-sizing: content-box; display: inline; font-family: MathJax_Main; font-size: 12.3018px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">0</span><span style="border: 0px; box-sizing: content-box; display: inline-block; font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; height: 4.026em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-47" style="border: 0px; box-sizing: content-box; display: inline; font-family: MathJax_Main; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">,</span><span class="msubsup" id="MathJax-Span-48" style="border: 0px; box-sizing: content-box; display: inline; font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.175em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: content-box; display: inline-block; font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.037em;"><span style="border: 0px; box-sizing: content-box; clip: rect(3.394em, 1000.58em, 4.198em, -999.997em); font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.02em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-49" style="border: 0px; box-sizing: content-box; display: inline; font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-50" style="border: 0px; box-sizing: content-box; display: inline; font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-51" style="border: 0px; box-sizing: content-box; display: inline; font-family: MathJax_Main-bold; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">x</span></span></span><span style="border: 0px; box-sizing: content-box; display: inline-block; font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; height: 4.026em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: content-box; font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; left: 0.635em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.848em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-52" style="border: 0px; box-sizing: content-box; display: inline; font-family: MathJax_Main; font-size: 12.3018px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">1</span><span style="border: 0px; box-sizing: content-box; display: inline-block; font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; height: 4.026em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-53" style="border: 0px; box-sizing: content-box; display: inline; font-family: MathJax_Main; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">,</span><span class="mo" id="MathJax-Span-54" style="border: 0px; box-sizing: content-box; display: inline; font-family: MathJax_Main; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.175em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">.</span><span class="mo" id="MathJax-Span-55" style="border: 0px; box-sizing: content-box; display: inline; font-family: MathJax_Main; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.175em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">.</span><span class="mo" id="MathJax-Span-56" style="border: 0px; box-sizing: content-box; display: inline; font-family: MathJax_Main; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.175em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">.</span><span class="mo" id="MathJax-Span-57" style="border: 0px; box-sizing: content-box; display: inline; font-family: MathJax_Main; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.175em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">,</span><span class="msubsup" id="MathJax-Span-58" style="border: 0px; box-sizing: content-box; display: inline; font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.175em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: content-box; display: inline-block; font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 2.014em;"><span style="border: 0px; box-sizing: content-box; clip: rect(3.394em, 1000.58em, 4.198em, -999.997em); font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.02em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-59" style="border: 0px; box-sizing: content-box; display: inline; font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-60" style="border: 0px; box-sizing: content-box; display: inline; font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-61" style="border: 0px; box-sizing: content-box; display: inline; font-family: MathJax_Main-bold; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">x</span></span></span><span style="border: 0px; box-sizing: content-box; display: inline-block; font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; height: 4.026em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: content-box; font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; left: 0.635em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.848em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-62" style="border: 0px; box-sizing: content-box; display: inline; font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-63" style="border: 0px; box-sizing: content-box; display: inline; font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-64" style="border: 0px; box-sizing: content-box; display: inline; font-family: MathJax_Math-italic; font-size: 12.3018px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">n</span><span class="mo" id="MathJax-Span-65" style="border: 0px; box-sizing: content-box; display: inline; font-family: MathJax_Main; font-size: 12.3018px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">−</span><span class="mn" id="MathJax-Span-66" style="border: 0px; box-sizing: content-box; display: inline; font-family: MathJax_Main; font-size: 12.3018px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">1</span></span></span><span style="border: 0px; box-sizing: content-box; display: inline-block; font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; height: 4.026em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span></span><span style="border: 0px; box-sizing: content-box; display: inline-block; font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; height: 2.359em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; box-sizing: content-box; display: inline-block; font-family: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; height: 0.87em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.33em; width: 0px;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation" style="border: 0px; box-sizing: content-box; clip: rect(1px, 1px, 1px, 1px); display: inline; font-family: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; height: 1px; left: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; top: 0px; transition: none 0s ease 0s; user-select: none; vertical-align: 0px; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">x</mi></mrow><mn>0</mn></msub><mo>,</mo><msub><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">x</mi></mrow><mn>1</mn></msub><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><msub><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">x</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></math></span></span>. The algorithm uses a linear recurrence relation that generates:</p><p style="background-color: white; border: 0px; box-sizing: inherit; clear: both; color: #232629; font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", "Liberation Sans", sans-serif; font-size: 15px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin-bottom: var(--s-prose-spacing); margin-left: 0px; margin-right: 0px; margin-top: 0px; padding: 0px; vertical-align: baseline;"><span class="MathJax_Preview" color="inherit" style="background: inherit; border: 0px; box-sizing: inherit; font-family: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;"></span></p><div class="MathJax_Display" style="background-attachment: inherit; background-clip: inherit; background-image: inherit; background-origin: inherit; background-position: inherit; background-repeat: inherit; background-size: inherit; border: 0px; box-sizing: inherit; font-family: inherit; font-size: 15px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 1em 0em; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow: auto hidden; padding: 0px; position: relative; text-align: center; vertical-align: baseline; width: 659px;"><span class="MathJax" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">x</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>+</mo><mi>k</mi></mrow></msub><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">x</mi></mrow><mi>k</mi></msub><mo>,</mo><msub><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">x</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">x</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mi>k</mi><mo>+</mo><mi>m</mi></mrow></msub><mo>,</mo><mi>r</mi><mo>,</mo><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">A</mi></mrow><mo stretchy="false">)</mo><mo>,</mo></math>" id="MathJax-Element-14-Frame" role="presentation" style="border: 0px; box-sizing: inherit; direction: ltr; display: inline; float: none; font-family: inherit; font-stretch: inherit; font-variant: inherit; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; position: relative; vertical-align: baseline; white-space: nowrap; word-spacing: normal;" tabindex="0"><nobr aria-hidden="true" style="border: 0px; box-sizing: inherit; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-67" style="border: 0px; box-sizing: content-box; display: inline-block; font-family: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 15.463em;"><span style="border: 0px; box-sizing: content-box; display: inline-block; font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 13.336em;"><span style="border: 0px; box-sizing: content-box; clip: rect(1.44em, 1013.28em, 2.761em, -999.997em); font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.353em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-68" style="border: 0px; box-sizing: content-box; display: inline; font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-69" style="border: 0px; box-sizing: content-box; display: inline; font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: content-box; display: inline-block; font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 2.014em;"><span style="border: 0px; box-sizing: content-box; clip: rect(3.394em, 1000.58em, 4.198em, -999.997em); font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.02em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-70" style="border: 0px; box-sizing: content-box; display: inline; font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-71" style="border: 0px; box-sizing: content-box; display: inline; font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-72" style="border: 0px; box-sizing: content-box; display: inline; font-family: MathJax_Main-bold; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">x</span></span></span><span style="border: 0px; box-sizing: content-box; display: inline-block; font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; height: 4.026em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: content-box; font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; left: 0.635em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.848em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-73" style="border: 0px; box-sizing: content-box; display: inline; font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-74" style="border: 0px; box-sizing: content-box; display: inline; font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-75" style="border: 0px; box-sizing: content-box; display: inline; font-family: MathJax_Math-italic; font-size: 12.3018px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">n</span><span class="mo" id="MathJax-Span-76" style="border: 0px; box-sizing: content-box; display: inline; font-family: MathJax_Main; font-size: 12.3018px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">+</span><span class="mi" id="MathJax-Span-77" style="border: 0px; box-sizing: content-box; display: inline; font-family: MathJax_Math-italic; font-size: 12.3018px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">k</span></span></span><span style="border: 0px; box-sizing: content-box; display: inline-block; font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; height: 4.026em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-78" style="border: 0px; box-sizing: content-box; display: inline; font-family: MathJax_Main; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.29em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">=</span><span class="mi" id="MathJax-Span-79" style="border: 0px; box-sizing: content-box; display: inline; font-family: MathJax_Math-italic; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.29em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">f<span style="border: 0px; box-sizing: content-box; display: inline-block; font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; height: 1px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.06em;"></span></span><span class="mo" id="MathJax-Span-80" style="border: 0px; box-sizing: content-box; display: inline; font-family: MathJax_Main; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">(</span><span class="msubsup" id="MathJax-Span-81" style="border: 0px; box-sizing: content-box; display: inline; font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: content-box; display: inline-block; font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.037em;"><span style="border: 0px; box-sizing: content-box; clip: rect(3.394em, 1000.58em, 4.198em, -999.997em); font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.02em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-82" style="border: 0px; box-sizing: content-box; display: inline; font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-83" style="border: 0px; box-sizing: content-box; display: inline; font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-84" style="border: 0px; box-sizing: content-box; display: inline; font-family: MathJax_Main-bold; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">x</span></span></span><span style="border: 0px; box-sizing: content-box; display: inline-block; font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; height: 4.026em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: content-box; font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; left: 0.635em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.848em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-85" style="border: 0px; box-sizing: content-box; display: inline; font-family: MathJax_Math-italic; font-size: 12.3018px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">k</span><span style="border: 0px; box-sizing: content-box; display: inline-block; font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; height: 4.026em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-86" style="border: 0px; box-sizing: content-box; display: inline; font-family: MathJax_Main; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">,</span><span class="msubsup" id="MathJax-Span-87" style="border: 0px; box-sizing: content-box; display: inline; font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.175em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: content-box; display: inline-block; font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 1.957em;"><span style="border: 0px; box-sizing: content-box; clip: rect(3.394em, 1000.58em, 4.198em, -999.997em); font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.02em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-88" style="border: 0px; box-sizing: content-box; display: inline; font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-89" style="border: 0px; box-sizing: content-box; display: inline; font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-90" style="border: 0px; box-sizing: content-box; display: inline; font-family: MathJax_Main-bold; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">x</span></span></span><span style="border: 0px; box-sizing: content-box; display: inline-block; font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; height: 4.026em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: content-box; font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; left: 0.635em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.848em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-91" style="border: 0px; box-sizing: content-box; display: inline; font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-92" style="border: 0px; box-sizing: content-box; display: inline; font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-93" style="border: 0px; box-sizing: content-box; display: inline; font-family: MathJax_Math-italic; font-size: 12.3018px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">k</span><span class="mo" id="MathJax-Span-94" style="border: 0px; box-sizing: content-box; display: inline; font-family: MathJax_Main; font-size: 12.3018px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">+</span><span class="mn" id="MathJax-Span-95" style="border: 0px; box-sizing: content-box; display: inline; font-family: MathJax_Main; font-size: 12.3018px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">1</span></span></span><span style="border: 0px; box-sizing: content-box; display: inline-block; font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; height: 4.026em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-96" style="border: 0px; box-sizing: content-box; display: inline; font-family: MathJax_Main; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">,</span><span class="msubsup" id="MathJax-Span-97" style="border: 0px; box-sizing: content-box; display: inline; font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.175em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span style="border: 0px; box-sizing: content-box; display: inline-block; font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 2.244em;"><span style="border: 0px; box-sizing: content-box; clip: rect(3.394em, 1000.58em, 4.198em, -999.997em); font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.02em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-98" style="border: 0px; box-sizing: content-box; display: inline; font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-99" style="border: 0px; box-sizing: content-box; display: inline; font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-100" style="border: 0px; box-sizing: content-box; display: inline; font-family: MathJax_Main-bold; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">x</span></span></span><span style="border: 0px; box-sizing: content-box; display: inline-block; font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; height: 4.026em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; box-sizing: content-box; font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; left: 0.635em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.848em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-101" style="border: 0px; box-sizing: content-box; display: inline; font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-102" style="border: 0px; box-sizing: content-box; display: inline; font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-103" style="border: 0px; box-sizing: content-box; display: inline; font-family: MathJax_Math-italic; font-size: 12.3018px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">k</span><span class="mo" id="MathJax-Span-104" style="border: 0px; box-sizing: content-box; display: inline; font-family: MathJax_Main; font-size: 12.3018px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">+</span><span class="mi" id="MathJax-Span-105" style="border: 0px; box-sizing: content-box; display: inline; font-family: MathJax_Math-italic; font-size: 12.3018px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">m</span></span></span><span style="border: 0px; box-sizing: content-box; display: inline-block; font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; height: 4.026em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-106" style="border: 0px; box-sizing: content-box; display: inline; font-family: MathJax_Main; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">,</span><span class="mi" id="MathJax-Span-107" style="border: 0px; box-sizing: content-box; display: inline; font-family: MathJax_Math-italic; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.175em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">r</span><span class="mo" id="MathJax-Span-108" style="border: 0px; box-sizing: content-box; display: inline; font-family: MathJax_Main; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">,</span><span class="texatom" id="MathJax-Span-109" style="border: 0px; box-sizing: content-box; display: inline; font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.175em; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-110" style="border: 0px; box-sizing: content-box; display: inline; font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-111" style="border: 0px; box-sizing: content-box; display: inline; font-family: MathJax_Main-bold; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">A</span></span></span><span class="mo" id="MathJax-Span-112" style="border: 0px; box-sizing: content-box; display: inline; font-family: MathJax_Main; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">)</span><span class="mo" id="MathJax-Span-113" style="border: 0px; box-sizing: content-box; display: inline; font-family: MathJax_Main; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">,</span></span><span style="border: 0px; box-sizing: content-box; display: inline-block; font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; height: 2.359em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; box-sizing: content-box; display: inline-block; font-family: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; height: 1.27em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.33em; width: 0px;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation" style="border: 0px; box-sizing: content-box; clip: rect(1px, 1px, 1px, 1px); display: inline; font-family: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; height: 1px; left: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; top: 0px; transition: none 0s ease 0s; user-select: none; vertical-align: 0px; width: 231.938px;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">x</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>+</mo><mi>k</mi></mrow></msub><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">x</mi></mrow><mi>k</mi></msub><mo>,</mo><msub><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">x</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">x</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mi>k</mi><mo>+</mo><mi>m</mi></mrow></msub><mo>,</mo><mi>r</mi><mo>,</mo><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">A</mi></mrow><mo stretchy="false">)</mo><mo>,</mo></math></span></span></div><p></p><p style="background-color: white; border: 0px; box-sizing: inherit; clear: both; color: #232629; font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", "Liberation Sans", sans-serif; font-size: 15px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin-bottom: var(--s-prose-spacing); margin-left: 0px; margin-right: 0px; margin-top: 0px; padding: 0px; vertical-align: baseline;">where <span class="MathJax_Preview" color="inherit" style="background: inherit; border: 0px; box-sizing: inherit; font-family: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;"></span><span class="MathJax" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>&#x2A7D;</mo><mi>m</mi><mo>&#x2A7D;</mo><mi>n</mi></math>" id="MathJax-Element-15-Frame" role="presentation" style="border: 0px; box-sizing: inherit; direction: ltr; display: inline; float: none; font-family: inherit; font-stretch: inherit; font-variant: inherit; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; position: relative; vertical-align: baseline; white-space: nowrap; word-spacing: normal;" tabindex="0"><nobr aria-hidden="true" style="border: 0px; box-sizing: inherit; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-114" style="border: 0px; box-sizing: content-box; display: inline-block; font-family: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 5.463em;"><span style="border: 0px; box-sizing: content-box; display: inline-block; font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 4.716em;"><span style="border: 0px; box-sizing: content-box; clip: rect(1.67em, 1004.72em, 2.819em, -999.997em); font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.526em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-115" style="border: 0px; box-sizing: content-box; display: inline; font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mn" id="MathJax-Span-116" style="border: 0px; box-sizing: content-box; display: inline; font-family: MathJax_Main; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">1</span><span class="mo" id="MathJax-Span-117" style="border: 0px; box-sizing: content-box; display: inline; font-family: MathJax_AMS; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.29em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">⩽</span><span class="mi" id="MathJax-Span-118" style="border: 0px; box-sizing: content-box; display: inline; font-family: MathJax_Math-italic; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.29em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">m</span><span class="mo" id="MathJax-Span-119" style="border: 0px; box-sizing: content-box; display: inline; font-family: MathJax_AMS; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.29em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">⩽</span><span class="mi" id="MathJax-Span-120" style="border: 0px; box-sizing: content-box; display: inline; font-family: MathJax_Math-italic; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.29em; position: static; transition: none 0s ease 0s; vertical-align: 0px;">n</span></span><span style="border: 0px; box-sizing: content-box; display: inline-block; font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; height: 2.532em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; box-sizing: content-box; display: inline-block; font-family: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; height: 1.07em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.197em; width: 0px;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation" style="border: 0px; box-sizing: content-box; clip: rect(1px, 1px, 1px, 1px); display: inline; font-family: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; height: 1px; left: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; top: 0px; transition: none 0s ease 0s; user-select: none; vertical-align: 0px; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>⩽</mo><mi>m</mi><mo>⩽</mo><mi>n</mi></math></span></span> and <span class="MathJax_Preview" color="inherit" style="background: inherit; border: 0px; box-sizing: inherit; font-family: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;"></span><span class="MathJax" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math>" id="MathJax-Element-16-Frame" role="presentation" style="border: 0px; box-sizing: inherit; direction: ltr; display: inline; float: none; font-family: inherit; font-stretch: inherit; font-variant: inherit; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; position: relative; vertical-align: baseline; white-space: nowrap; word-spacing: normal;" tabindex="0"><nobr aria-hidden="true" style="border: 0px; box-sizing: inherit; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-121" style="border: 0px; box-sizing: content-box; display: inline-block; font-family: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0.578em;"><span style="border: 0px; box-sizing: content-box; display: inline-block; font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.463em;"><span style="border: 0px; box-sizing: content-box; clip: rect(1.67em, 1000.46em, 2.474em, -999.997em); font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.296em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-122" style="border: 0px; box-sizing: content-box; display: inline; font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-123" style="border: 0px; box-sizing: content-box; display: inline; font-family: MathJax_Math-italic; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">r</span></span><span style="border: 0px; box-sizing: content-box; display: inline-block; font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; height: 2.302em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; box-sizing: content-box; display: inline-block; font-family: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; height: 0.67em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.063em; width: 0px;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation" style="border: 0px; box-sizing: content-box; clip: rect(1px, 1px, 1px, 1px); display: inline; font-family: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; height: 1px; left: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; top: 0px; transition: none 0s ease 0s; user-select: none; vertical-align: 0px; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math></span></span> and <span class="MathJax_Preview" color="inherit" style="background: inherit; border: 0px; box-sizing: inherit; font-family: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;"></span><span class="MathJax" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">A</mi></mrow></math>" id="MathJax-Element-17-Frame" role="presentation" style="border: 0px; box-sizing: inherit; direction: ltr; display: inline; float: none; font-family: inherit; font-stretch: inherit; font-variant: inherit; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; overflow-wrap: normal; padding: 0px; position: relative; vertical-align: baseline; white-space: nowrap; word-spacing: normal;" tabindex="0"><nobr aria-hidden="true" style="border: 0px; box-sizing: inherit; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none 0s ease 0s; vertical-align: 0px;"><span class="math" id="MathJax-Span-124" style="border: 0px; box-sizing: content-box; display: inline-block; font-family: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 1.037em;"><span style="border: 0px; box-sizing: content-box; display: inline-block; font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none 0s ease 0s; vertical-align: 0px; width: 0.865em;"><span style="border: 0px; box-sizing: content-box; clip: rect(1.497em, 1000.81em, 2.532em, -999.997em); font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.353em; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-125" style="border: 0px; box-sizing: content-box; display: inline; font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-126" style="border: 0px; box-sizing: content-box; display: inline; font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-127" style="border: 0px; box-sizing: content-box; display: inline; font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;"><span class="mi" id="MathJax-Span-128" style="border: 0px; box-sizing: content-box; display: inline; font-family: MathJax_Main-bold; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px;">A</span></span></span></span><span style="border: 0px; box-sizing: content-box; display: inline-block; font-family: inherit; font-size: 17.4px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; height: 2.359em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; box-sizing: content-box; display: inline-block; font-family: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; height: 0.937em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none 0s ease 0s; vertical-align: -0.063em; width: 0px;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation" style="border: 0px; box-sizing: content-box; clip: rect(1px, 1px, 1px, 1px); display: inline; font-family: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; height: 1px; left: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; top: 0px; transition: none 0s ease 0s; user-select: none; vertical-align: 0px; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">A</mi></mrow></math></span></span> are objects that can be specified as parameters in the algorithm. Since the seed gives the initial set of vectors (and given other fixed parameters for the algorithm), the series of pseudo-random numbers generated by the algorithm is fixed. If you change the seed then you change the initial vectors, which changes the pseudo-random numbers generated by the algorithm. This is, of course, the function of the seed.</p><p style="background-color: white; border: 0px; box-sizing: inherit; clear: both; color: #232629; font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", "Liberation Sans", sans-serif; font-size: 15px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin-bottom: var(--s-prose-spacing); margin-left: 0px; margin-right: 0px; margin-top: 0px; padding: 0px; vertical-align: baseline;">Now, it is important to note that this is just one example, using the MT19937 algorithm. There are many PRNGs that can be used in statistical software, and they each involve different recursive methods, and so the seed means a different thing (in technical terms) in each of them. You can find a library of PRNGs for <code style="background-color: var(--black-075); border-radius: 3px; border: 0px; box-sizing: inherit; color: var(--black-800); font-family: var(--ff-mono); font-size: 13px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; padding: 2px 4px; vertical-align: baseline; white-space: pre-wrap;">R</code> in <a href="http://stat.ethz.ch/R-manual/R-devel/library/base/html/Random.html" rel="noreferrer" style="border: 0px; box-sizing: inherit; cursor: pointer; font-family: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; padding: 0px; user-select: auto; vertical-align: baseline;">this documentation</a>, which lists the available algorithms and the papers that describe these algorithms.</p><p style="background-color: white; border: 0px; box-sizing: inherit; clear: both; color: #232629; font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", "Liberation Sans", sans-serif; font-size: 15px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">The purpose of the seed is to allow the user to "lock" the pseudo-random number generator, to allow replicable analysis. Some analysts like to set the seed using a <a href="https://en.wikipedia.org/wiki/Hardware_random_number_generator" rel="noreferrer" style="border: 0px; box-sizing: inherit; cursor: pointer; font-family: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; padding: 0px; user-select: auto; vertical-align: baseline;">true random-number generator (TRNG)</a> which uses hardware inputs to generate an initial seed number, and then report this as a locked number. If the seed is set and reported by the original user then an auditor can repeat the analysis and obtain the same sequence of pseudo-random numbers as the original user. If the seed is not set then the algorithm will usually use some kind of default seed (e.g., from the system clock), and it will generally not be possible to replicate the randomisation.</p><p style="background-color: white; border: 0px; box-sizing: inherit; clear: both; color: #232629; font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", "Liberation Sans", sans-serif; font-size: 15px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;"><br /></p><p style="background-color: white; border: 0px; box-sizing: inherit; clear: both; color: #232629; font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", "Liberation Sans", sans-serif; font-size: 15px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Using a <u>Quantum random number</u> source allows for much greater security as we can create numbers that are irreducibly random. Using a quantum-entanglement photon source based on the <u>Beta-Barium Borate non-linear crystal</u>, we can configure photo detectors that can detect the H and V modes of pair of entangled 810nm photons, These photons will be in superposition until the moment they are detected and so will represent the perfect 50/50 coin toss. </p><p style="background-color: white; border: 0px; box-sizing: inherit; clear: both; color: #232629; font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", "Liberation Sans", sans-serif; font-size: 15px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;"><br /></p><p style="background-color: white; border: 0px; box-sizing: inherit; clear: both; color: #232629; font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", "Liberation Sans", sans-serif; font-size: 15px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">The setup is showcased in the following video:</p><div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen="" class="BLOG_video_class" height="266" src="https://www.youtube.com/embed/6RGLiOdyLfo" width="320" youtube-src-id="6RGLiOdyLfo"></iframe></div><br /><p style="background-color: white; border: 0px; box-sizing: inherit; clear: both; color: #232629; font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", "Liberation Sans", sans-serif; font-size: 15px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;"><br /></p></div><h1 style="background-color: white; color: #202122; font-family: sans-serif; margin: 0.5em 0px; text-align: left;"><u><span style="font-size: large;"><br /></span></u></h1><h1 style="background-color: white; color: #202122; font-family: sans-serif; margin: 0.5em 0px; text-align: left;"><u><span style="font-size: large;">Entangled Photon Imaging</span></u></h1><div><u><span style="font-size: large;"><br /></span></u></div><div><span style="font-size: medium;">The diagram below showcases the scheme for quantum encryption using images of entangled photons processed using bitwise XOR on any image we want to encrypt using a shared quantum key distrubted between 2 users, Alice and Bob. </span></div><div><span style="font-size: medium;"><br /></span></div><div><span style="font-size: medium;">Each recipient recieves the relative anti-correlations to each other and so can encrypt or decrypt their transferred image in a one time pad use.</span></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEgoCWstcKAFIAuu3_evytPlRqUN3rFlMhkxdmGw2BL0T_JdBKVm8uoz3tW1ydNMxaIGrFhGEY8YCGXj2zEqHbqCWbpmsHKacMZJKx0Mi7ih2c1Yhy6wzLGRbj4Ykx6csedkNGeHFTvEiEbEOKsCAQzETom7i44EdD0x1bxn9ppAe63HvOuB_dVv7CtzFA=s555" style="margin-left: 1em; margin-right: 1em;"><img alt="XOR Quantum Encryption Using Entangled Photons" border="0" data-original-height="389" data-original-width="555" height="448" src="https://blogger.googleusercontent.com/img/a/AVvXsEgoCWstcKAFIAuu3_evytPlRqUN3rFlMhkxdmGw2BL0T_JdBKVm8uoz3tW1ydNMxaIGrFhGEY8YCGXj2zEqHbqCWbpmsHKacMZJKx0Mi7ih2c1Yhy6wzLGRbj4Ykx6csedkNGeHFTvEiEbEOKsCAQzETom7i44EdD0x1bxn9ppAe63HvOuB_dVv7CtzFA=w640-h448" width="640" /></a></div><br /><div>using a CCD with a large enough sensor we can image pairs of entangled photons at 810nm using a narrow single-band pass filter at around 800-830nm</div><div><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEgLZMKQLbBEE894IKIHFEt0aDEFUnhsOCwtH5nPwpLZdhLtNSjznkiiL_EFeRbxgk6u7K_VNhfobVi6ChOmxeZjfKil0auQ7A1QMRLfITSl7Gdyt3o_Tt5f5V1pQaAkxRk39YYl0KxffHJV7CAFJMYhcSCJEuOGu4jH-Bx3V0o6h9GtGdTFgtpT0HvYqg=s1600" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1200" data-original-width="1600" height="240" src="https://blogger.googleusercontent.com/img/a/AVvXsEgLZMKQLbBEE894IKIHFEt0aDEFUnhsOCwtH5nPwpLZdhLtNSjznkiiL_EFeRbxgk6u7K_VNhfobVi6ChOmxeZjfKil0auQ7A1QMRLfITSl7Gdyt3o_Tt5f5V1pQaAkxRk39YYl0KxffHJV7CAFJMYhcSCJEuOGu4jH-Bx3V0o6h9GtGdTFgtpT0HvYqg=s320" width="320" /></a></div><br /><div><br /></div><h1 style="background-color: white; color: #202122; font-family: sans-serif; margin: 0.5em 0px; text-align: left;"><u><span style="font-size: large;">Quantum image encryption and decryption</span></u></h1><p style="background-color: white; margin: 0.5em 0px;"><span face="sans-serif" style="color: #202122;"><span style="font-size: 14px;"><br /></span></span></p><p style="background-color: white; margin: 0.5em 0px;"><span style="color: #202122; font-family: arial;">Images contain information in both the form of metadata on the conditions the image was captured and in the pixel data itself. Metadata can be encrypted as a data stream so we will refer to image encryption as encryption of the pixel information itself from this point.</span></p><p style="background-color: white; margin: 0.5em 0px;"><span style="color: #202122; font-family: arial;">The definition of image encryption is an extension of data encryption in general: through the bitwise XOR operation of the original image pixels and the key image pixels, with the key image being either a pseudo-random stream cipher or the quantum random stream cipher or anti-correlated entangled information shared over a secure channel.</span></p><p style="background-color: white; margin: 0.5em 0px;"><span style="color: #202122; font-family: arial;"><br /></span></p><p style="background-color: white; margin: 0.5em 0px;"><span style="color: #202122; font-family: arial;">In either case we Perform bitwise XOR operation on the encrypted image and the key image.</span></p><p style="background-color: white; margin: 0.5em 0px;"><span style="color: #202122; font-family: arial;"><br /></span></p><p style="background-color: white; margin: 0.5em 0px;"><span style="color: #202122; font-family: arial;">It can be seen from the image encryption and decryption that they are all the same operation.</span></p><p style="background-color: white; margin: 0.5em 0px;"><span style="color: #202122; font-family: arial;"><br /></span></p><p style="background-color: white; margin: 0.5em 0px;"><span style="color: #202122; font-family: arial;">We now stipulate that the literal symbol of XOR is xor. According to the above bitwise XOR operation, we assume:</span></p><p style="background-color: white; margin: 0.5em 0px;"><span style="color: #202122; font-family: arial;"><br /></span></p><p style="background-color: white; margin: 0.5em 0px;"><span style="color: #202122; font-family: arial;">xor(a,b)=c</span></p><p style="background-color: white; margin: 0.5em 0px;"><span style="color: #202122; font-family: arial;"><br /></span></p><p style="background-color: white; margin: 0.5em 0px;"><span style="color: #202122; font-family: arial;">You can get:</span></p><p style="background-color: white; margin: 0.5em 0px;"><span style="color: #202122; font-family: arial;"><br /></span></p><p style="background-color: white; margin: 0.5em 0px;"><span style="color: #202122; font-family: arial;">xor(c,b)=a</span></p><p style="background-color: white; margin: 0.5em 0px;"><span style="color: #202122; font-family: arial;"><br /></span></p><p style="background-color: white; margin: 0.5em 0px;"><span style="color: #202122; font-family: arial;">Or:</span></p><p style="background-color: white; margin: 0.5em 0px;"><span style="color: #202122; font-family: arial;"><br /></span></p><p style="background-color: white; margin: 0.5em 0px;"><span style="color: #202122; font-family: arial;">xor(c,a)=b</span></p><p style="background-color: white; margin: 0.5em 0px;"><span style="color: #202122; font-family: arial;"><br /></span></p><p style="background-color: white; margin: 0.5em 0px;"><span style="color: #202122; font-family: arial;">In summary, we assume that a is the original image data and b is the key, then c calculated by xor(a,c) is the encrypted ciphertext. </span></p><p style="background-color: white; margin: 0.5em 0px;"><span style="color: #202122; font-family: arial;">In summary:</span></p><p style="background-color: white; margin: 0.5em 0px;"><span style="color: #202122; font-family: arial;"><br /></span></p><p style="background-color: white; margin: 0.5em 0px;"><span style="color: #202122; font-family: arial;">Encryption process: Perform a bitwise XOR operation on the image a and the key b to complete the encryption and obtain the ciphertext c.</span></p><p style="background-color: white; margin: 0.5em 0px;"><span style="color: #202122; font-family: arial;"><br /></span></p><p style="background-color: white; margin: 0.5em 0px;"><span style="color: #202122; font-family: arial;">Decryption process: Perform a bitwise XOR operation on the ciphertext c and the key b, complete the decryption, and get the image a.</span></p><p style="background-color: white; color: #202122; margin: 0.5em 0px;"><span style="font-family: arial;"><br /></span></p><p style="background-color: white; color: #202122; margin: 0.5em 0px;"><span style="font-family: arial;"><br /></span></p><p style="background-color: white; color: #202122; margin: 0.5em 0px;"><span style="font-family: arial;">We can use our quantum random numbers generator in 2 ways to create our image encryption key:</span></p><p style="background-color: white; color: #202122; margin: 0.5em 0px;"><span style="font-family: arial;"><br /></span></p><p style="background-color: white; color: #202122; margin: 0.5em 0px;"><span style="font-family: arial;">(1) as a random number generator seed source</span></p><p style="background-color: white; color: #202122; margin: 0.5em 0px;"><span style="font-family: arial;">(2) using the random superposition of the H and V modes</span></p><p style="background-color: white; color: #202122; margin: 0.5em 0px;"><span style="font-family: arial;"><br /></span></p><p style="background-color: white; color: #202122; margin: 0.5em 0px;"><span style="font-family: arial;">We can also use the shared set of correlated images, captured using the single CCD, from our entangled photon source with Alice getting one half and Bob getting the anti-correlated half. This provides the perfect key, with the quantum images shared over a separate channel hidden from the encrypted classical images.</span></p><p style="background-color: white; color: #202122; margin: 0.5em 0px;"><span style="font-family: arial;"><br /></span></p><p style="background-color: white; color: #202122; margin: 0.5em 0px;"><span style="font-family: arial;">The file exchange channel is 2 way:</span></p><p style="background-color: white; color: #202122; margin: 0.5em 0px;"><span style="font-family: arial;"><br /></span></p><p style="background-color: white; color: #202122; margin: 0.5em 0px;"><span style="font-family: arial;">Alice can use Her key to encrypt the image, Bob can use His key to decrypt the image</span></p><p style="background-color: white; color: #202122; margin: 0.5em 0px;"><span style="font-family: arial;">OR</span></p><p style="background-color: white; color: #202122; margin: 0.5em 0px;"><span style="font-family: arial;">Bob can use His key to encrypt the image, Alice can use Her key to decrypt the image.</span></p><p style="background-color: white; color: #202122; margin: 0.5em 0px;"><span style="font-family: arial;"><br /></span></p><p style="background-color: white; color: #202122; margin: 0.5em 0px;"><span style="font-family: arial;">The XOR Cipher in this use can also be extended as a component in more complex overlay network ciphers if need be however for computational efficiency it is not necessary. It is just as effectual to have 1 quantum cipher as many, so in effect the system is completely hidden, by virtue of hidden variables, and is encrypted in an information condensate.</span></p><p style="background-color: white; color: #202122; margin: 0.5em 0px;"><span style="font-family: arial;"><br /></span></p><p style="background-color: white; color: #202122; margin: 0.5em 0px;"><span style="font-family: arial;">We can use my suite of image analysis plots to perform histogram analysis and construct a correlogram to try and do a routine check to see if the image encryption could be vulnerable to frequency analysis from the encrypted image pixel data:</span></p><p style="background-color: white; color: #202122; margin: 0.5em 0px;"><span style="font-family: arial;"><br /></span></p><h1 style="background-color: white; color: #202122; margin: 0.5em 0px; text-align: left;"><u><span style="font-family: arial; font-size: medium;">Afterward:</span></u></h1><p style="background-color: white; color: #202122; margin: 0.5em 0px;"><span style="font-family: arial;"><br /></span></p><p style="background-color: white; color: #202122; margin: 0.5em 0px;"><span style="font-family: arial;">Encryption of information, in particular visual information in the form of images and video, is of ever increasing importance online. Encryption of image metadata can be accomplished in tandem with encryption of pixel data to make for a more robust as well as computational inexpensive process using the XOR cipher combined with quantum RNGs and anti-correlated quantum information generated via entanglement sources. </span></p><p style="background-color: white; color: #202122; margin: 0.5em 0px;"><span style="font-family: arial;">Research is also ongoing to use a variety of different entanglement sources, both optical (using Non-linear crystals) and in the Microwave Domain (using Josephson Junctions) with the goal being a many spectrum approach to secure information transfer. </span></p><p style="background-color: white; color: #202122; margin: 0.5em 0px;"><span style="font-family: arial;">This would allow, among other things a secure, quantum entangled channel transferring the anti-correlated entangled key to operate on a separate band from the wavelengths used to generate the entangled key for instance. </span></p><p style="background-color: white; color: #202122; margin: 0.5em 0px;"><span style="font-family: arial;">Moreover although line-of-sight optical communication has had much fanfare it sees that the microwave band still offers superior connectivity in telecoms infrastructure and will most likely be the focus of further generations of secure data transfer in mobile applications. Quantum Encryption therefore is a key niche in this area and XOR-based ciphers will be the fastest to implement in terms of wireless/non-fiber secure quantum communication. </span></p><p style="background-color: white; color: #202122; margin: 0.5em 0px;"><span style="font-family: arial;"> </span></p><h1 style="background-color: white; color: #202122; margin: 0.5em 0px; text-align: left;"><span style="font-family: arial; font-size: small;">Github Files:</span></h1><h1 style="background-color: white; color: #202122; font-family: sans-serif; margin: 0.5em 0px; text-align: left;"><br /></h1><h1 style="background-color: white; color: #202122; font-family: sans-serif; margin: 0.5em 0px; text-align: left;">https://github.com/MuonRay/Quantum-Encryption-of-Images-using-Bitwise-XOR-and-QRNG</h1><p style="background-color: white; color: #202122; font-family: sans-serif; margin: 0.5em 0px;"><br /></p>MuonRayhttp://www.blogger.com/profile/03712859045968965104noreply@blogger.com0tag:blogger.com,1999:blog-5996268492156364201.post-87810722159265876542021-12-17T03:35:00.003-08:002021-12-17T03:55:53.098-08:00Drone Archaeology | Tomb Hunting Using NDVI, NIR Threshold with Bitwise NOT in Python<iframe frameborder="0" height="270" src="https://youtube.com/embed/iGgtG44AXw4" style="background-image: url(https://i.ytimg.com/vi/iGgtG44AXw4/hqdefault.jpg);" width="480"></iframe><div><br /></div><div><span face="Roboto, Noto, sans-serif" style="background-color: white; color: #0d0d0d; font-size: 15px; white-space: pre-wrap;">Coding Repository:</span><span face="Roboto, Noto, sans-serif" style="background-color: white; color: #0d0d0d; font-size: 15px; white-space: pre-wrap;"><a href="https://github.com/MuonRay/Drone_Archaeology">https://github.com/MuonRay/Drone_Archaeology</a></span><span face="Roboto, Noto, sans-serif" style="background-color: white; color: #0d0d0d; font-size: 15px; white-space: pre-wrap;">
Here I showcase recent developments I have made in using Near-Infrared imaging with drones in order to perform feature detection in the environment, in this case for archaeological examination. Python coding was used to process drone-captured Near-Infrared Images into Normalised Differential Vegetation Index (NDVI) greyscale images which are further processed using both a segmentation of ndvi around the tomb region followed by a contour overlay in the perimeter of the tombs.
The lower values of NIR reflectance can be caused by plant growth stress, itself potentially caused by partially or completely submerged rock from a tomb, wall, cairn or road. The image processing technique makes use of the bitwise xor function to highlight the lower regions of NIR reflectance, segmenting them using a threshold mask. By highlighting the low NIR regions and performing NDVI on these regions, we can create a clear image feature in which we can draw a clear contour around using an automated contour tracer.
Version 2 uses a standard contour, Version 4 is an attempt, with limited success, to overlay feature boxes over the tomb images with the intent to extend the code into more automated feature detection of individual boulders, cairn formations and other features.
Updates to this last step will be ongoing, with hopes to combine with other indexes, such as ENDVI, to allow for better segmentation of ancient structures that may create vegetation stress that can be sensed remotely. Drones specifically designed to use NDVI can make use of more calibrated NIR captures that can provide greater accuracy however we have found the modified DJI Mavic Pro 2 offers unparalleled image resolution, having a 20MP camera as compared to more standard 5MP NDVI cameras on the market.</span></div>MuonRayhttp://www.blogger.com/profile/03712859045968965104noreply@blogger.com0tag:blogger.com,1999:blog-5996268492156364201.post-76314574400828922882021-12-03T23:16:00.001-08:002021-12-03T23:16:03.224-08:00Quantum Entanglement Cryptography - Technical Development and Notes<div dir="ltr" style="text-align: left;" trbidi="on">
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As theories go, quantum mechanics has certainly been successful. Despite its many counter-intuitive predictions, it has provided an accurate description of the atomic world for more than 80 years. It has also been an essential tool for designing today's computer chips and hard-disk drives, as well as the lasers used in the fibre-optic communications of the Internet. Now, however, the ability to manipulate the quantum states of individual subatomic particles is allowing us to exploit the strange properties of quantum theory much more directly in information technology.</div>
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We are used to thinking of information as being abstract, but in fact all information requires a physical medium for its processing, storage and communication. The basic unit of information - a bit that is either "0" or "1" - can be represented physically by, for example, the current in a circuit or light in an optical fibre. As information is represented by ever smaller physical systems, quantum effects become increasingly important. The ultimate limit comes when bits are represented by the quantum state of a single particle, such as the polarization of a photon.</div>
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Applied to information, quantum theory throws up some very odd predictions. These are not only interesting as a test of quantum mechanics, but can also bring us practical applications that are simply impossible with "classical" information technology. </div>
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<span style="font-size: large;"><u>QKD - Quantum Key Distribution </u></span></h2>
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<span face="Arial, Helvetica, sans-serif">Overview of Classical Cryptography</span></h3>
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<span face="Arial, Helvetica, sans-serif">Cryptography is a vital part of today's computer and communication networks, protecting everything from business e-mails to bank transactions and Internet shopping. Information is generally kept secret using a mathematical formula called an encryption algorithm, together with a secret "key" that the sender uses to scramble a message into a form that cannot be understood by an eavesdropper. The recipient then uses the same key - typically a long binary number - with a decryption algorithm to read the message.</span></div>
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<span face="Arial, Helvetica, sans-serif"><span style="line-height: 17.92px;">Encryption usually involves one or more secret keys—numbers used in some mathematical operation to protect the sensitive information. For example, suppose the </span><span style="color: red; line-height: 17.92px;">message to be sent is the number 4</span><span style="line-height: 17.92px;">, the</span><span style="color: blue; line-height: 17.92px;"> key is the number 3</span><span style="line-height: 17.92px;">, and the encryption scheme is simple multiplication. Then the</span><span style="color: lime; line-height: 17.92px;"> encrypted message is 12</span><span style="line-height: 17.92px;"> (because </span><span style="color: red; line-height: 17.92px;">4</span><span style="line-height: 17.92px;"> x </span><span style="color: blue; line-height: 17.92px;">3</span><span style="line-height: 17.92px;"> = </span><span style="color: lime; line-height: 17.92px;">12</span><span style="line-height: 17.92px;">). The receiver would divide the transmitted number, 12, by the key, 3, to recover the original number, 4. In this example, the procedure (or <span style="color: purple;">algorithm</span>) to generate the encrypted message from the message sent was too simple. There are a finite number of multiples that we can search that will generate 12,so we could have easily cracked the encryption.</span></span><br />
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<span face="Arial, Helvetica, sans-serif"><span style="line-height: 17.92px;">Ideally, you want the key to be a number which has the least number of divisible elements, which is a prime number, since our <span style="color: blue;">key was 3</span> this is a candidate. This key is small, but remember even the most powerful computers can't have infinitely large keys. For our message we must pass it through an algorithm in such a way so that we get he most use out of our key. In another example, imagine the </span><span style="color: red; line-height: 17.92px;">message sent was 50</span><span style="line-height: 17.92px;">, the <span style="color: purple;">algorithm is a one-way function that divides the message by 10, and adds 4</span> to this new number generating <span style="color: red;">9</span>. We can now encrypt this message with our key, generating <span style="color: red;">9</span> x <span style="color: blue;">3</span> =<span style="color: lime;"> 27</span>.</span></span><br />
<span face="Arial, Helvetica, sans-serif"><span style="line-height: 17.92px;">Our <span style="color: lime;">encrypted message is 27</span>. If a person had intercepted the <span style="color: lime;">27</span> as the message, without the algorithm or the key they would be at a loss to get <span style="color: red;">50</span> by mathematical guesswork alone. </span></span><br />
<span face="Arial, Helvetica, sans-serif"><span style="line-height: 17.92px;">Algorithms can be kept in the public domain, hence in a</span></span><span face="Arial, Helvetica, sans-serif" style="line-height: 17.92px;"> more realistic application, the secret key would have to be hundreds of digits long to be used in an algorithm effectively.</span><br />
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<span face="Arial, Helvetica, sans-serif"><span style="line-height: 17.92px;">There are 2 broad classes of algorithm – <span style="color: red;">symmetric</span> and <span style="color: blue;">asymmetric</span>. Symmetric algorithms use the same key for both encryption and decryption. Asymmetric algorithms, such as public/private key cryptography use one key for encryption and a different, though mathematically related key for decryption. </span></span><br />
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<span face="Arial, Helvetica, sans-serif"><span style="line-height: 17.92px;">This sounds somewhat counter-intuitive. So too does the idea that keeping algorithms in the public domain still maintains secrecy. Nevertheless, In cryptography it’s not the algorithm that is needed to be kept secret. The algorithm should be designed in such a way that if it is discovered, unless the hacker has the key, Key secrecy is what’s important. Its even fair to say that if you know the algorithm inside and out, such that , you know what mathematics was used and you can reverse engineer the procedure done on the ciphertext itself, a good encryption algorithm will still keep the plaintext data secret by virtue of key security. A lock is only as good as its key in this field, a somewhat counter-intuitive notion indeed.</span></span><br />
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<span face="Arial, Helvetica, sans-serif" style="line-height: 17.92px;">Almost the entirety of public/private key cryptography (used by protocols such as SSL/TLS) is based on the notion that there is no pattern to a series of prime numbers, other than that they are prime.</span><br />
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<span face="Arial, Helvetica, sans-serif"><span style="line-height: 17.92px;">There is a possibility that somebody has already come up with a prime-prediction algorithm. It would certainly be in their interest to keep it secret!</span></span><br />
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<span face="Arial, Helvetica, sans-serif"><span style="line-height: 17.92px;">The encryption algorithm has 2 inputs – plaintext and the key. It has one output, ciphertext.</span></span><br />
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<span face="Arial, Helvetica, sans-serif"><span style="line-height: 17.92px;">If decrypting data, 2 inputs: ciphertext and the key. It has one output, plaintext.</span></span><br />
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<span face="Arial, Helvetica, sans-serif" style="line-height: 17.92px;">such that the key </span><span face="Arial, Helvetica, sans-serif" style="line-height: 17.92px;">is a prime number, only dvionly much more sophisticated than simple multiplication. Either way, without the key, you can't unlock the information.</span><br />
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<span face="Arial, Helvetica, sans-serif"><span style="line-height: 17.92px;">One Way Functions</span></span></h3>
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<span face="Arial, Helvetica, sans-serif"><span style="line-height: 17.92px;">Mathematical functions where it is difficult (or impossible) to get back to the source values, knowing only the output values, are known as one-way functions. There are many, but modular arithmetic gives us a method that is used extensively in cryptography.</span></span><br />
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<span face="Arial, Helvetica, sans-serif"><span style="line-height: 17.92px;">A simple example is where, on a 12 hour clock face, you add 5 hours to 9am. The answer is 2 pm. Or written down we could say:</span></span><br />
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<span face="Arial, Helvetica, sans-serif"><span style="line-height: 17.92px;">9+5=2</span></span><br />
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<span face="Arial, Helvetica, sans-serif"><span style="line-height: 17.92px;">Because this is an example of modular arithmetic where the modulus is 12, we’d actually write:</span></span><br />
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<span face="Arial, Helvetica, sans-serif"><span style="line-height: 17.92px;">9+5=2(mod12)</span></span><br />
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<span face="Arial, Helvetica, sans-serif"><span style="line-height: 17.92px;">Let’s take a simple function:</span></span><br />
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<span style="font-size: large;"><b><span face="'Segoe UI', 'Lucida Grande', Verdana, Arial, Helvetica, sans-serif" style="background-color: #ced5db; color: #333333; line-height: 18.1656px;">3</span><sup style="background-color: #ced5db; color: #333333; font-family: "Segoe UI", "Lucida Grande", Verdana, Arial, Helvetica, sans-serif; line-height: 18.1656px;">x </sup><span face="'Segoe UI', 'Lucida Grande', Verdana, Arial, Helvetica, sans-serif" style="background-color: #ced5db; color: #333333; line-height: 18.1656px;">where x=2</span></b></span><br />
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<span face="Arial, Helvetica, sans-serif"><span style="line-height: 17.92px;">This is a function for turning 2 in to 9, because it’s the same as 3 * 3, which equals 9. There is a direct relationship between the magnitude of x and the magnitude of the function result. Using modular arithmetic can give the function a great property – <span style="color: red;">unpredictability </span>and/or <span style="color: #2b00fe;">randomness</span></span></span><br />
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<tr><td valign="top" width="94"><span style="font-size: large;"><b>x</b></span></td><td valign="top" width="41"><span style="font-size: large;"><b>1</b></span></td><td valign="top" width="36"><span style="font-size: large;"><b>2</b></span></td><td valign="top" width="57"><span style="font-size: large;"><b>3</b></span></td><td valign="top" width="57"><span style="font-size: large;"><b>4</b></span></td><td valign="top" width="57"><span style="font-size: large;"><b>5</b></span></td><td valign="top" width="57"><span style="font-size: large;"><b>6</b></span></td></tr>
<tr><td valign="top" width="94"><span style="font-size: large;"><b>3<sup>x</sup></b></span></td><td valign="top" width="41"><span style="font-size: large;"><b>3</b></span></td><td valign="top" width="36"><span style="font-size: large;"><b>9</b></span></td><td valign="top" width="57"><span style="font-size: large;"><b>27</b></span></td><td valign="top" width="57"><span style="font-size: large;"><b>81</b></span></td><td valign="top" width="57"><span style="font-size: large;"><b>243</b></span></td><td valign="top" width="57"><span style="font-size: large;"><b>729</b></span></td></tr>
<tr><td valign="top" width="94"><span style="font-size: large;"><b>3<sup>x</sup>(mod7)</b></span></td><td valign="top" width="41"><span style="font-size: large;"><b>3</b></span></td><td valign="top" width="36"><span style="font-size: large;"><b>2</b></span></td><td valign="top" width="57"><span style="font-size: large;"><b>6</b></span></td><td valign="top" width="57"><span style="font-size: large;"><b>4</b></span></td><td valign="top" width="57"><span style="font-size: large;"><b>5</b></span></td><td valign="top" width="57"><span style="font-size: large;"><b>1</b></span></td></tr>
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<span face="Arial, Helvetica, sans-serif"><span style="line-height: 17.92px;">Many computer programs, such as the computer desktop for example, used password in Window stored as a one way function – albeit one that is considerably more complex than what you’ve just seen.</span></span><br />
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<h3 style="background-color: #ced5db; clear: both; color: #260859; font-family: "Segoe UI Semibold", "Segoe UI", "Lucida Grande", Verdana, Arial, Helvetica, sans-serif; font-size: 1.45em; font-weight: normal; line-height: 21.9501px; margin: 3px 0px;">
Classical Key Distribution</h3>
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The problem with encrypting/decrypting data by <span style="color: #800180;"><u>symmetric</u></span> means is that you have to somehow get the decryption key safely to your partner. This makes it possible to be intercepted if transmitted over a public channel, such as the internet. It’s the modern equivalent of an age-old problem that generals have had of communicating with their officers in the field for centuries. If the messenger who has the key is captured, all your communication can be decrypted, whether or not subsequent messengers know the key. </div>
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<span style="line-height: 1.4em;">Essentially, using the one-way function alone in encryption protocols requires no authentication, so either side could be be spoofed by an active wiretapper. </span><span style="line-height: 1.4em;">This is called the key distribution problem.</span></div>
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3 mathematicians came up with an answer to this problem: Diffie, Hellman and Merkle. They do an exchange of data which can be intercepted by anybody but which allows both sender and receiver to generate the same key but doesn’t allow the interceptor to generate the key. </div>
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The Diffie-Hellman Key Exchange allows two principals to agree on a shared key even though they exchange messages in public.</div>
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<span style="line-height: 1.4em;">Using modular arithmetic it is possible to derive an algorithm that demonstrates how to perform the exchange between Alice and Bob.</span></div>
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<span style="font-size: 15px; line-height: 21px;">The protocol can easily be extended into one that does also implement the necessary authentication.</span></div>
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<span style="font-size: 15px; line-height: 21px;">The first step is to choose a large prime number p (around 512 bits). The second is to choose an integer g where g < p (with some other technical restrictions.) The protocol works as follows:</span></div>
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<span style="font-size: 15px; line-height: 21px;">At this point, A can compute:</span></div>
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<span style="font-size: 15px; line-height: 21px;">(TB)SA </span></div>
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<span style="font-size: 15px; line-height: 21px;">= (gSB mod p)SA </span></div>
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<span style="font-size: 15px; line-height: 21px;">= (gSB)SA mod p </span></div>
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<span style="font-size: 15px; line-height: 21px;">= ((gSBSA) mod p).</span></div>
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<span style="font-size: 15px; line-height: 21px;">Similarly B can compute :</span></div>
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<span style="font-size: 15px; line-height: 21px;">(TA)SB </span></div>
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<span style="font-size: 15px; line-height: 21px;">= </span><span style="font-size: 15px; line-height: 1.4em;">(gSA mod p)SB</span></div>
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<span style="font-size: 15px; line-height: 21px;">= (gSA)SB mod p </span></div>
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<span style="font-size: 15px; line-height: 21px;">= ((gSASB) mod p).</span></div>
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<span style="font-size: 15px; line-height: 21px;">Therefore, ((gSASB) mod p) = ((gSBSA) mod p) is the final shared key.</span></div>
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Follow the steps 1 through 4. In the last step both Alice and Bob have the same key: 9. From this point on they can use 9 as their universal encryption and decryption key. <br />
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<span style="font-size: 15px; line-height: 21px;">A wiretapper can see all the messages that are sent, but can't do anything without having a fast way to compute logs in finite fields, which is assumed to be hard. One problem with Diffie-Hellman is that it does not generalize to send arbitrary messages.</span></div>
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<span style="font-size: 15px; line-height: 21px;">Physical Analogy for Diffie-Hellman Key Exchange</span></div>
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<span style="font-size: 15px; line-height: 21px;">We can use a physical analogy to better understand the principles underlining Diffie-Hellman key exchange. Consider the following:</span></div>
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<span style="font-size: 15px; line-height: 21px;">We have two principals, A and B, each with a 3-liter paint pot that contains 1-liter of yellow paint. We will use E to denote a passive wiretapper. We can assume that mixed paint cannot be deconstructed into original colors.</span></div>
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<span style="font-size: 15px; line-height: 21px;">A adds to her 1 liter of yellow paint a secret color SA. B also adds to his 1 liter of yellow paint a secret color SB.</span></div>
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<span style="font-size: 15px; line-height: 21px;">A and B swap pots. E is able to observe the 2, 2-liter mixtures be exchanged, but E cannot deduce what color was added to either mixture, E can only deduce the relative color balance in the combined 4 liter mixture: 2 * yellow + SA + SB (Y:Y:SA:SB).</span></div>
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<span style="font-size: 15px; line-height: 21px;">A adds SA to B's pot. The result (Y:SA:SB) is the key. B adds SB to A's pot. The result (Y:SB:SA) is the key.</span></div>
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<span style="font-size: 15px; line-height: 21px;">Notice: A and B have computed the same key, but E gets a different one.</span></div>
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Public Key Cryptography</h3>
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In public key cryptography, some keys are known to everyone, so it would seem that the key distribution problem vanishes. The approach was first published in the open literature in 1975. (Recently declassified documents in Great Britain suggest that public key cryptography was known there before 1975.)</div>
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The basic idea of a public key cryptosystem is to have two keys: a private (secret) key and a public key. Anyone can know the public key. Plaintext to a principal B is encrypted using B's public key. B decrypts the enciphered text using its private key. As long as B is the only one who knows the private key, then only B can decrypt messages encrypted under B's public key.</div>
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Some public key cryptography schemes also allow plaintext to be run through the decryption algorithm (using the private key). What is produced is referred to as <em>signed text</em> and it can be "deciphered" using the public key. Only the possessor of a private key can create text that is decipherable using the public key. The functionality of signed text cannot be replicated using secret key/symmetric cryptography.</div>
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Public key cryptography is usually <em>much</em> slower than secret key cryptography, so it is rarely used to encrypt an entire message. Typically a message is encrypted using shared key cryptography (with a secret key). That secret key is then encrypted using public key cryptography, and the encrypted message and key are sent. This is known as <em>hybrid encryption</em>. This method can allow for complex structures in implementing our secrecy requirements (see Figure below) : e.g. "message is readable by A,B,C,D".<br />
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History of Public Key Cryptography</h4>
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<em>(United States)</em></div>
<ul style="font-size: medium; line-height: normal;">
<li>1975: Diffie imagines asymmetric cryptography (Diffie + Hellman)</li>
<li>1976: Diffie-Hellman key exchange</li>
<li>April 1977: RSA (Rivest, Shamir, Adelman)</li>
</ul>
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<em>(United Kingdom)</em></div>
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<li>1969: Government Communications Headquarters (GCHQ) - succesor to Bletchly Park - asks James Ellis to look into the key distribution problem. Ellis recalls a Bell Labs report about adding noise to a signal, transmitting it, and then removing the noise.</li>
<li>1973: Clifford Cocks (recent Cambridge Math Ph.D) joins GCHQ. He hears about Ellis idea and searches for a suitable function, and he thinks of RSA. GCHQ now could do public key encryption.</li>
<li>January 1974: Malcolm Williamson, in an effort to try to break Cock's work, discovers Diffie-Hellman.</li>
</ul>
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Uses of Public-Key Cryptography</h3>
<span style="background-color: white; font-size: small; line-height: normal;">Uses of public key cryptography include secrecy, authentication, and digital signatures.</span><br />
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Secrecy is obtained when principal A encrypts a message m using B's public key. Thereafter, the only way to decrypt m is to know the private key of B. (see Figure below)</div>
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<img height="56" src="http://www.cs.cornell.edu/courses/cs513/2005fa/L26_secrecy.gif" width="472" /></center>
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In secret key cryptography, doing authentication requires having a different key for each pair of principals; in public key cryptography, each principal needs to know just its own private key. An example of a public-key authentication protocol is:</div>
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<img height="276" src="http://www.cs.cornell.edu/courses/cs513/2005fa/L26_authen.gif" width="441" /></center>
<span style="background-color: white; font-size: small; line-height: normal;">Digital signatures are used to prove that a message was generated by a particular principal. Assume that the cryptosystem has the additional property wherein a message m "decrypted" under a private key, and then "encrypted" using the corresponding public key produces m. To create a signed message, A will encrypt a message using its own private key and send that encrypted message to B. B looks up A's public key and uses it to decrypt the message. This is not completely practical since it requires running the decryption on an entire message, which can be expensive. A solution is to compute a hash of the message and sign that.</span><br />
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A hash is a function that digests information. It takes a message as input and outputs a short bit string (say, 128 bits). An example of a 1-bit hash would be a function that returns the parity of the message. Think of a hash as a succint summary of a message that has four properties:</div>
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<li>It is <em>computationally infeasible</em> to determine the input message m based on the digest of that message hash(m), which means the digest must convey no information about the original message.</li>
<li>It is infeasible to find any message with a given digest value, which means we can't attack by replacing a message m1 with another message m2 with the same hash value.</li>
<li>It is infeasible to find 2 messages with a given hash. If we don't have this property, then it is possible a person could sign a message, then the signature could be cut and pasted on to another message with the same hash.</li>
<li>And finally, changing even 1-bit of the input gets completely different output, so that syntactically similar messages generate very different outputs and it is not likely that two bit-strings with the same hash value could be mistaken for each other.</li>
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These properties make a message-text substitution attack difficult given a hash. Specifically, suppose that message m is sent along with a signed hash value for m. The properties of the hash function would make it difficult for an attacker to substitute another meaningful message that has the same hash value as the original.</div>
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We can easily have multiple signatures (see Figure 1 below), as well as build up a chain of signatures which establishes a valid history (see Figure 2 below). This chaining of signatures can be used to prove such a claim as "<i>Alice had signed the message when I got it.</i>".</div>
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Examples of Public-Key Cryptosystems</h3>
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We will now examine a few examples of public key cryptography systems.</div>
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Merkle's Puzzles</h4>
<em style="font-size: medium; line-height: normal;">Merkle's Puzzles</em><span style="background-color: white; font-size: small; line-height: normal;"> was one of the first public key cryptographic systems to be described. It allows A and B to agree on a secret key. Principal A invents a million keys and a million puzzles, where each puzzle encodes a different one of the keys. Each puzzle is assumed to take at least two minutes to solve and fit into 96 bits. A sends these puzzles to B. B then picks a puzzle at random and solves it. B encrypts a pre-arranged string (say 0000) with the key from the puzzle it solved. B sends this encrypted string back to A. A trys each of the million keys on the message it receives from B. The one that decrypts the message and obtains the pre-arranged string is the secret key that A will use henceforth to communicate with B.</span><br />
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A wiretapper C could steal the million puzzles. However, C would need to crack all million of the puzzles in order to discover the secret key. (If the wiretapper didn't know the pre-arranged string, then it can't even use a known-plaintext attack.) Since cracking each puzzle requires at least 2 minutes, the wiretapper would need on average 330 days to find the key.</div>
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Certification Authorities (Public Key Infrastructure)</h3>
<span style="background-color: white; font-size: small; line-height: normal;">It would seem that an advantage to public key cryptography is that a KDC is no longer necessary. However, how can one principal learn the public key another? How does one principal know they have the </span><em style="font-size: medium; line-height: normal;">right</em><span style="background-color: white; font-size: small; line-height: normal;"> public key and haven't been spoofed by an intruder? It turns out that some sort of server is still needed to certify which public keys belong to whom.</span><br />
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A <em>certification authority</em> (CA) is a trusted server that generates certificates of the form {name, public key}<sub>CA</sub> where CA is the certification authority's signature (private) key. All hosts are preconfigured with the certification authority's public key, therefore any host can check the signature on these certificates. Note that a CA is more attractive than a KDC because a CA it doesn't need to be on-line. Certificates can be stored anyplace and forwarded anywhere as they are needed.</div>
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Asymmetric Key Encryption</h3>
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We use one key to encrypt, and a related key to decrypt data. You can actually swop the keys round. But the point is you don’t have one key. This gets round the key distribution problem. There’s a great way of describing the difference between symmetric and asymmetric key encryption. It involves the use of a box to put messages in and we have to assume the box, its clasps and the padlock used to lock it are impossible to penetrate.</div>
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Symmetric Key: You send a messenger out with a copy of the key. He gets it to your recipient who lives 10 miles away. On the way he stops at a pub and has his pocket picked. The key is whisked off to a locksmith who copies it and it is then secreted back in to the messenger’s pocket.</div>
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Some time later you send a messenger with the box containing your message. You are confident that your recipient is the only one who can read the message because the original messenger returned and reported nothing unusual about the key. The second messenger stops at the same pub. He is pick-pocketed. The copy key is used to unlock the box and read the message. The box with its message intact is secreted back in to the messenger’s pocket. You and your recipient have no idea that your communication has been compromised. There is no secrecy…</div>
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Asymmetric Key: Your recipient has a padlock and key. He keeps the key in a private place about his person. Let’s therefore call it a private key. He puts the padlock in to the box, but leaves it unlocked. He doesn’t mind if anybody sees the padlock. It’s publicly viewable. Even though it’s not <em>really</em> a key, let’s call it a public key. He sends a messenger to you with the box. The messenger stops at the pub and is pick pocketed. All the snoopers see is an open padlock. They secretly return the box. The messenger arrives at your door. You take the padlock out of the box and put your message in to it. You use the open padlock to lock the box, snapping it shut and you send the messenger on his way. He again stops at the pub and is pick-pocketed. They find only a padlocked box. No key. They have no way of getting in to the box. They secretly return the box to the messenger’s pocket. The messenger gets to your recipient, who use the key he secreted in a private place about his person (the private key) and uses it to unlock the padlock and read the message. Secrecy is maintained.</div>
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You can see the process is a bit more complicated for asymmetric key than for symmetric key, so it’s not something you’d want to do often. So what is often done is that instead of putting a message in the box and padlocking it, a symmetric key is put in the box and padlocked. That way, you solve the key distribution problem. That’s what happens with computer cryptography mostly. Public/private key cryptography is used to transport a symmetric key that is used for message exchanges. One reason for doing this is that asymmetric key crypto, or public/private key crypto, as it is known, is expensive, in terms of computing power, whereas symmetric key crypto is much more lightweight. When you see that a web site uses 256 bit encryption, they are talking about the symmetric key that is used<em> after</em> the public/private key crypto was used to transport the symmetric key from sender to receiver. Often the key lengths for public/private key cryptography is 2048 bits. You may have found yourself confused when setting up IIS with 256 bit SSL encryption and seeing keys of 1024 or 2048 bits. This is why – it’s the difference between what’s called the <em>session key</em> and the public/private keys.</div>
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Although the diagram above explains how 2 keys are used, where does all this public and private key malarkey come in to play?</div>
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Let’s take the example of an ecommerce web server that wants to provide SSL support so you can send your credit card details securely over the Internet. Look at the public and private keys in the following diagram.</div>
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<a href="http://blogs.msdn.com/cfs-file.ashx/__key/CommunityServer-Blogs-Components-WeblogFiles/00-00-01-40-06-metablogapi/5824.image_5F00_37025BC0.png" style="color: #0066dd; cursor: pointer; outline: none; text-decoration: none;"><img alt="image" border="0" height="481" src="http://blogs.msdn.com/cfs-file.ashx/__key/CommunityServer-Blogs-Components-WeblogFiles/00-00-01-40-06-metablogapi/1538.image_5F00_thumb_5F00_5BB37337.png" style="border-width: 0px; display: block; float: none; height: auto; margin-left: auto; margin-right: auto; max-width: 100%; overflow: hidden;" title="image" width="640" /></a>The public and private keys are held on the ecommerce web server. The private key is heavily protected in the keystore. Many organisations will go as far as to have a special tamper-proof hardware device to protect their private keys. The public key doesn’t need to be protected because it’s, well, public. You could have daily printouts of it in the newspapers and have it broadcast every hour, on the hour, on the radio. The idea is that it doesn’t matter who sees it.</div>
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The website generates the public and private keys. They have to be generated as a <em>key-pair</em> because they are mathematically related to each other. You retrieve the public key from the website and use it as your encryption key. You’re not just going to send your credit card information across the Internet yet. You’re actually going to generate a symmetric key and that is going to become the plain-text input data to the asymmetric encryption algorithm. The cipher-text will traverse the Internet and the ecommerce site will now use its private key to decrypt the data. The resulting output plain-text will be the symmetric key you sent. Now that both you and the ecommerce site have a symmetric key that was transported secretly, you can encrypt all the data you exchange. This is what happens with a URL that starts <em>https://</em>.</div>
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There are still a couple of problems to solve here, but let’s put them on to the back-burner for a little while. We need to understand digital signatures and certificates for those problems. In the meantime let’s have a peek at the mathematics inside the public/private key algorithm. There is an interesting little story-ette around this algorithm. A researcher at the UK’s <a href="http://www.gchq.gov.uk/" style="color: #0066dd; cursor: pointer; outline: none; text-decoration: none;" target="_blank">GCHQ</a> called <a href="http://en.wikipedia.org/wiki/Clifford_Cocks" style="color: #0066dd; cursor: pointer; outline: none; text-decoration: none;" target="_blank">Clifford Cocks</a> invented the algorithm in 1973. However, working for GCHQ, his work was secret, so he couldn’t tell anybody.<br />
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About 3 years later, 3 mathematicians, Ron Rivest, Adi Shamir and Leonard Adelman also invented it. They went on to create the security company RSA (which stands for Rivest, Shamir and Adelman). It is said the RSA algorithm is the most widely used piece of software in the world.<br />
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RSA Algorithm</h4>
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<span style="background-color: white; font-size: small; line-height: normal;">RSA is usually used to encrypt a private key and then send that with along with a message encrypted </span><em style="font-size: medium; line-height: normal;">by</em><span style="background-color: white; font-size: small; line-height: normal;"> the private key. It uses a variable key length (usually 512 bits) and a variable block size that is not greater than the key length. RSA works as follows:</span></div>
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<li>Choose two large primes (say, 256 bits each) p and q. These must be kept secret.</li>
<li>Compute n = p*q. The number n is not secret. This systems works under the assumption that factoring n is computationally intractable.</li>
<li>Chose e such that e is relatively prime to (has no common factors other than 1 with) (p-1)*(q-1). The number e is usually chosen to be small. 3 and 64437 are popular.</li>
<li>The public key is the pair (e, n). Note that e doesn't have to be secret. The private key is (d, n) where d is the multiplicative inverse of e mod (p-1)(q-1).</li>
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<span style="background-color: white; font-size: small; line-height: normal;">To encrypt a message m, compute m</span><sup style="line-height: normal;">e</sup><span style="background-color: white; font-size: small; line-height: normal;"> mod n and send the result as ciphertext. To decrypt ciphertext c: m = c</span><sup style="line-height: normal;">d</sup><span style="background-color: white; font-size: small; line-height: normal;"> mod n. RSA can also be used for digital signatures. To sign a message m: s = m</span><sup style="line-height: normal;">d</sup><span style="background-color: white; font-size: small; line-height: normal;"> mod n. To check a signature: m = s</span><sup style="line-height: normal;">e</sup><span style="background-color: white; font-size: small; line-height: normal;"> mod n.</span></div>
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A lot of number theory is needed to prove that this technique works. One necessary theorem is: m = (m<sup>e</sup> mod n)<sup>d</sup> mod n.</div>
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First, we’ll generate the public key. We pick 2 random giant prime numbers. In this case, I’ll pick 2 small primes to keep it simple; 17 and 11. We multiply them to get 187. We then pick another prime; 7. That’s our public key – 2 numbers. Pretty simple.</div>
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Now we use the public key to generate the private key. We run it through the algorithm in the diagram above. You can see we use modular arithmetic. Obviously the numbers would be massive in real life. But here, we end up with a private key of 23. The function, 7 * d = 1(mod 160) has that look of simplicity, but it’s not like that at all. With large numbers we’d need to use the <a href="http://en.wikipedia.org/wiki/Extended_Euclidean_algorithm" style="color: #0066dd; cursor: pointer; outline: none; text-decoration: none;">Extended Euclidean Algorithm</a>. I have to say, my eyes glazed over and I was found staring in to the distance when I read this about it:</div>
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<em>The extended Euclidean algorithm is particularly useful when a and b are </em><a href="http://en.wikipedia.org/wiki/Coprime" style="color: #0066dd; cursor: pointer; outline: none; text-decoration: none;"><em>coprime</em></a><em>, since x is the </em><a href="http://en.wikipedia.org/wiki/Modular_multiplicative_inverse" style="color: #0066dd; cursor: pointer; outline: none; text-decoration: none;"><em>modular multiplicative inverse</em></a><em> of a </em><a href="http://en.wikipedia.org/wiki/Modular_arithmetic" style="color: #0066dd; cursor: pointer; outline: none; text-decoration: none;"><em>modulo</em></a><em> b.</em></div>
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Now we want to use that to encrypt a message.</div>
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To keep things simple, we’ll send a single character; “X”. ASCII for X is 88. As we are the sender, we only know the public key’s 2 values: 187 and 7, or N and e. Running 88 through the simple algorithm gives us the value 11. We send the ciphertext value 11 to the ecommerce web server.</div>
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The Web server has access to the private key, so it can decrypt the ciphertext.</div>
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The web server passes the plaintext through the algorithm shown above and gets us the original “X” that was sent. The bit that says:</div>
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Plaintext = 11<sup>23</sup>(mod 187)</div>
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OK – there’s actually a problem here. In this message, every “X” would come out in ciphertext as the value 11. We could perform a <a href="http://en.wikipedia.org/wiki/Frequency_analysis" style="color: #0066dd; cursor: pointer; outline: none; text-decoration: none;" target="_blank">frequency analysis</a> attack on the message. In the English language, certain letters tend to appear more frequently than others. The letters “e” and “i” for example are very common, but “x” and “z” are uncommon.<a href="http://blogs.msdn.com/cfs-file.ashx/__key/CommunityServer-Blogs-Components-WeblogFiles/00-00-01-40-06-metablogapi/8032.Englishslf1_5F00_6854621D.png" style="color: #0066dd; cursor: pointer; outline: none; text-decoration: none;"><img alt="English-slf[1]" border="0" height="432" src="http://blogs.msdn.com/cfs-file.ashx/__key/CommunityServer-Blogs-Components-WeblogFiles/00-00-01-40-06-metablogapi/7065.Englishslf1_5F00_thumb_5F00_5382699D.png" style="border: 0px; display: block; float: none; height: auto; margin-left: auto; margin-right: auto; max-width: 100%; overflow: hidden;" title="English-slf[1]" width="536" /></a></div>
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There is a “signature” that could be used to find the content of a message. We therefore need to encrypt much larger blocks of data than just one byte at a time.</div>
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Digital Signatures</h3>
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Asymmetric keys, as mentioned earlier can be swopped around. If you use one key for encryption, you must use the other key for decryption. This feature comes in very handy for the creation of digital signatures. You’ve heard of digitally signed documents, authenticode, digitally signed applications, digital certificates and so on.</div>
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In the diagram you can see all we’ve done is combined some plaintext in to the same “message” as its equivalent ciphertext. When it’s time to check a digital signature, we reverse the process:</div>
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To check a message, we decrypt the encrypted portion, and get back plain text. We then compare that to the plain text in the message. If the 2 sets of plain text are different, it means either:</div>
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<li>The plaintext in the message has been altered and that accounts for the difference.</li>
<li>The ciphertext in the message has been altered and that accounts for the difference.</li>
<li>They have both been altered and that accounts for the difference.</li>
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In order to have a consistent message, the attacker would need to have access to the key that was used to generate the ciphertext.</div>
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Do you remember earlier, I talked about hashes? Well, because a message might be quite large, it’s often best to generate a hash of the message and encrypt that. If it’s an MD5 hash, it means you’ll only have to encrypt 128 bytes. When you come to perform the validation of the signature, you have to take the plain text portion and generate a hash before you do the comparison. It just uses the CPU more efficiently.</div>
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In this case, the message consists of a small section of ciphertext because the string-size of the input plaintext was reduced through hashing before it was encrypted. It also includes the plaintext of the message.</div>
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Depending on the data you are looking at, you’ll often even find the keys you can use to decrypt the message in plaintext within the message body. It seems like complete madness because anybody who intercepts the message could simply modify the plaintext portion of the message and then use the included key to generate a new ciphertext equivalent. That would make the message consistent.</div>
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However, if the plaintext key included in the message is the message-issuer’s public key, then the attacker would need access to the corresponding private key, which they won’t get because it’s, well, private.</div>
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But even with this there is still a problem. How do you know the message came from the sender it purports to come from? As an attacker, I could easily generate my own key-pair. I could then create a message that says I am the issuer and use my private key to create the encrypted part of the message.</div>
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When you come to check the message you’ll know that it definitely wasn’t tampered with in transit, but how do you know you can trust the public key embedded in to the message? How do you know that it’s me that created the message. That’s where digital certificates come in to play.<br />
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Certificates</h3>
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Certificates are data structures that conform to a specification: X.509. But really they are just documents that do what we just talked about. The plain text data is the public key, plus other distinguishing information like the issuer, the subject name, common name and so on. It is then hashed and the hash is encrypted using the private key of a special service called a certification authority (CA) – a service that issues certificates.</div>
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When we protect a web server with an SSL certificate, we go through a 2 stage process. generating a certificate request, and then finishing it off by receiving and installing the certificate. The request part, generates a public and private key. The public key plus the distinguishing information is sent to the CA, which then creates a digitally signed document, signed using the CA’s private key. The document conforms to X.509 certificate standards. The certificate is returned by the CA, and we install it on our web server.</div>
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Anytime anybody connects to the web server over SSL, they retrieve the certificate and perform a signature validation on it. Remember it was signed by the CA’s private key. So they have to have the CA’s public key to perform the validation. If you go in to Internet Explorer’s Internet Options and then to the Content tab, you’ll se a Certificates button. That shows you all the CAs you have certificates (and therefore public keys) for. It means if you see a certificate that was signed by a CA on a web site, in theory, the CA did a check to make sure the requester was indeed the requester before issuing the certificate. It means that you have to trust that the CA did a good job of checking the requester’s validity before issuing the certificate.</div>
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Even this creates a minor problemette – how do you know the CA’s certificate wasn’t created by an imposter of some description? Well, it can have its certificate signed by a CA higher up the food-chain than itself. Eventually you get to a CA at the top of the food chain and this is called a Root CA. Internet Explorer and all the other browsers have all the main Root CAs for the Internet built-in. These are trusted organisations. They have to be! They are so trusted, they are able to sign their own certificates.</div>
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You may from time to time play with a Visual Studio command line tool called makecert.exe. It’s a tool that creates self-signed certificates. If you are just using them for development purposes on a local machine they are probably fine. You trust yourself, presumably. Sometimes you can use self-signed certificates on Internet-facing services. For example if you upload your own self-signed certificate to a service and you are sure nobody intercepted it while you were uploading it (because you were using SSL maybe), it means you can have private conversations with the service and you can be sure the service is the service you issued the certificate to. If you just sent the naked certificate, they’d be able to encrypt messages that only you could decrypt, because you’d have the private key. It’s possible to also include the private key when you create a certificate. If you send one of these certificates to an Internet service, they can digitally sign messages they send to you with your private key. Because you are assured that you gave the private key only to the service, you can be sure the messages are genuinely coming from that service and not an imposter. You have to trust that the service do a good job of keeping your private key safe.</div>
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Of course it wouldn’t be practical if every time you wanted to buy something on the Internet, in order to create an SSL connection you had to first upload a self-signed certificate. That’s why there is a large infrastructure of CAs and Root CAs built on the Internet. This infrastrucutre is called a Public Key Infrastructure or PKI. Many organisations have their own internal PKIs.</div>
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Above: Internet Explorer’s list of Trusted Root CAs.</div>
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You can also see the chain of CAs up to that chain’s corresponding Root CA when you look at a certificate.</div>
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<a href="http://blogs.msdn.com/cfs-file.ashx/__key/CommunityServer-Blogs-Components-WeblogFiles/00-00-01-40-06-metablogapi/6471.image_5F00_7248B7AE.png" style="color: #0066dd; cursor: pointer; outline: none; text-decoration: none;"><img alt="image" border="0" height="600" src="http://blogs.msdn.com/cfs-file.ashx/__key/CommunityServer-Blogs-Components-WeblogFiles/00-00-01-40-06-metablogapi/2843.image_5F00_thumb_5F00_005780DF.png" style="border-width: 0px; display: block; float: none; height: auto; margin-left: auto; margin-right: auto; max-width: 100%; overflow: hidden;" title="image" width="458" /></a>This shows an expired certificate that was issued to my BPOS (Office 365) account by the CA called “Microsoft Online Svcs BPOS EMEA CA1”. Its certificate was in turn issued by “Microsoft Services PCA” which had its certificate issued by “Microsoft Root Certificate Authority”. As it’s a Root CA, it appears in the Trusted Root CAs container in Internet Explorer. As you walk up the chain you have to eventually get to a point where you trust the certificate. If you don’t, you’ll get a certificate error warning and a lot of messages advising you not to continue. </div>
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I’ll write another post soon that goes through a complete SSL handshake. That's a great way to explain what’s happening in crypto.</div>
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<span style="line-height: 1.4em;"><span face="Helvetica, Verdana, Arial, sans-serif">The Necessity of Security - BB84 Quantum Cryptography Protocol</span></span></h3>
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<span style="line-height: 1.4em;"><span face="Helvetica, Verdana, Arial, sans-serif">Although modern algorithms such </span><span face="Arial, Helvetica, sans-serif">as the Advanced Encryption Standard (AES) are very hard to break without the key, this system suffers from an obvious weakness: the key must be known to both parties. Thus the problem of confidential communication reduces to that of how to distribute these keys securely - the encrypted me</span><span face="Helvetica, Verdana, Arial, sans-serif">ssage itself can then safely be sent along a public channel (figure 1). A common method is to use a trusted courier to transport the key from sender to receiver.</span></span></div>
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<tr><td align="center" class="caption"><span style="color: black;">Alice wishes to send Bob a secret message - say, a bank transaction - over a potentially insecure communication channel. To do this, Alice and Bob must share a secret key - a long binary number. Alice can then encrypt her message into "cipher text" using the key in conjunction with an encryption algorithm, such as AES. The cipher text may then be transmitted using an ordinary data channel, as it will be unintelligible to an eavesdropper, and Bob can use the key to decrypt the message. In contrast to traditional methods of key distribution, such as a trusted courier, quantum cryptography guarantees the secrecy of the key. The key can also be frequently changed, thereby reducing the risk of it being stolen or of it being deduced by crypto-analysis - statistical analysis of the cipher text.</span></td></tr>
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However, any distribution method that relies on humans is vulnerable to the key being revealed voluntarily or under coercion. In contrast, quantum cryptography, or more accurately quantum key distribution (QKD), provides an automated method for distributing secret keys using standard communication fibres. The revolutionary feature of QKD is that it is inherently secure: assuming that we study and correct for all the loopholes in the laws of quantum theory, we can prove that the key cannot be obtained by an eavesdropper without the sender and recipient's knowledge. Furthermore, QKD allows the key to be changed frequently, reducing the threat of key theft or "cryptanalysis", whereby an eavesdropper analyses patterns in the encrypted messages in order to deduce the secret key.<br />
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The first method for distributing secret keys encoded in quantum states was proposed in 1984 by theoretical physicists Charles Bennett at IBM and Gilles Brassard at the University of Montreal. In their "BB84" protocol, a bit of information is represented by the polarization state of a single photon - "0" by horizontal and "1" by vertical, for example. The sender (Alice) transmits a string of polarized single photons to the receiver (Bob) and by carrying out a series of quantum measurements and public communications they are able to establish a shared key and to test whether an eavesdropper (Eve) has intercepted any bits of this key en route.</div>
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<span face="Helvetica, Verdana, Arial, sans-serif"><span style="font-size: 15px; line-height: 21px;">In the Dirac notation, the polarization state <b>|H1,2></b> denotes a horizontal, and<b> |V1,2></b> a vertical polarization of photon 1 and 2, respectively. The superposition states</span></span><br />
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<span face="Helvetica, Verdana, Arial, sans-serif" style="font-size: large;"><span style="line-height: 21px;">|+> = 1/√2 (|H> + |V>) </span></span><br />
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<span face="Helvetica, Verdana, Arial, sans-serif" style="font-size: large;"><span style="line-height: 21px;">|->= 1/√2 (|H> - |V>)</span></span><br />
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<span face="Helvetica, Verdana, Arial, sans-serif"><span style="font-size: 15px; line-height: 21px;">correspond to +45° and -45° diagonal linear polarizations of the photons.</span></span><br />
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<span face="Helvetica, Verdana, Arial, sans-serif"><span style="font-size: 15px; line-height: 21px;">A prominent entangled quantum state of a photon pair is the so-called singlet state, which in this notation can be written as</span></span><br />
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<span face="Helvetica, Verdana, Arial, sans-serif" style="font-size: large;"><span style="line-height: 21px;">|Ψ-> = 1/√2 (|H1V2> - |V1H2>)</span></span><br />
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<span face="Helvetica, Verdana, Arial, sans-serif"><span style="font-size: 15px; line-height: 21px;">In a measurement which distinguishes horizontal and vertical polarization, this state would either lead to a result with photon 1 in horizontal and photon 2 in vertical, or with photon 1 in vertical and photon 2 in horizontal polarization: The measurement results are always opposite on both sides.</span></span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgKzKBRW9_1EogqJnK_8xm5fNZCFVzOgpTbpHy499a5eDnSNkhfH43lAm_dxalRnw91mF9W6_lIDEx4DwoCS23QXceW3OPlv5zeeRB3qGnr-WyjnRiIn9uYN1eWtKP2752vDxDn-vmTKyJU/s1600/pairidea.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="136" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgKzKBRW9_1EogqJnK_8xm5fNZCFVzOgpTbpHy499a5eDnSNkhfH43lAm_dxalRnw91mF9W6_lIDEx4DwoCS23QXceW3OPlv5zeeRB3qGnr-WyjnRiIn9uYN1eWtKP2752vDxDn-vmTKyJU/s1600/pairidea.png" width="640" /></a></div>
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<span face="Helvetica, Verdana, Arial, sans-serif"><span style="font-size: 15px; line-height: 21px;">This also holds if a measurement is carried out that distinguishes ±45° diagonal polarizations, since</span></span><br />
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<span face="Helvetica, Verdana, Arial, sans-serif"><span style="line-height: 21px;">both H/V and ±45° measurements, the results on both sides are anti-correlated.</span></span><br />
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<span face="Helvetica, Verdana, Arial, sans-serif"><span style="line-height: 21px;">In a real experiment, one would get either one of two possible outcomes on each side, for example H or V on one side, and + or - on the other side if a measurement apparatus distinguishes ±45° polarizations. For measurements on many pairs, one would get a number of events NH,+, NV,+, NH,-, and NV,- for all possible combinations. From this, one can define a correlation function</span></span><br />
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<i>E<sub>HV,±</sub> := (N<sub>H,+</sub> + N<sub>V,-</sub> - N<sub>H,-</sub> - N<sub>V,+</sub>) / N<sub>T</sub>,</i></div>
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<span face="Helvetica, Verdana, Arial, sans-serif" style="line-height: 21px;">with a total event number </span> <i>N<sub>T</sub> = N<sub>H,+</sub> + N<sub>V,-</sub> + N<sub>V,-</sub> + N<sub>H,-</sub></i> .</div>
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<span face="Helvetica, Verdana, Arial, sans-serif"><span style="line-height: 21px;">This correlation function can only take values between between -1 and +1. For the singlet state and the same measurements on both sides, the anti-correlation quantitatively reads </span></span><br />
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<i style="background-color: white; color: #333333; font-family: verdana, arial, helvetica, sans-serif; font-size: 16px;"> E<sub>HV,HV</sub> = E<sub>±,±</sub> =</i><span face="verdana, arial, helvetica, sans-serif" style="background-color: white; color: #333333; font-size: 16px;"> -1.</span><span face="Helvetica, Verdana, Arial, sans-serif"><span style="line-height: 21px;"><br /></span></span><br />
<span face="Helvetica, Verdana, Arial, sans-serif"><span style="line-height: 21px;">A correlation function can not only be defined for measurements H/V or ±45°, but for an arbitrary rotation angle φ with respect to the H/V directions.</span></span><br />
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<span face="Helvetica, Verdana, Arial, sans-serif"><span style="line-height: 21px;">The quantitative version of the Bell inequality now makes two choices of measurement orientations on the two sides; we refer to them as<span style="color: blue;"> a</span> and <span style="color: lime;">a'</span> on one side, and <span style="color: red;">b</span> and <span style="color: purple;">b' </span>on the other side. Correlation functions corresponding to these orientations ("settings") are combined to a new quantity:</span></span><br />
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<span face="Helvetica, Verdana, Arial, sans-serif"><span style="line-height: 21px;">With the argument of J.S. Bell, under the assumption that there is a local realistic model with "hidden" parameters determining the measurement outcomes, this quantity is bounded - expressed by the inequality </span></span><br />
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<span face="Helvetica, Verdana, Arial, sans-serif" style="line-height: 21px;"> </span><span face="verdana, arial, helvetica, sans-serif" style="background-color: white; color: #333333; font-size: 16px; text-align: center;">|</span><i style="background-color: white; color: #333333; font-family: verdana, arial, helvetica, sans-serif; font-size: 16px; text-align: center;">S</i><span face="verdana, arial, helvetica, sans-serif" style="background-color: white; color: #333333; font-size: 16px; text-align: center;">| ≤ 2</span><br />
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<span face="Helvetica, Verdana, Arial, sans-serif"><span style="line-height: 21px;">Quantum physics, on the other side, allows to calculate the expectation values of these correlation functions for a given state of the photon pairs. For a choice of settings <span style="color: blue;">a=H/V</span>, <span style="color: lime;">a'=±45°</span>, and settings<span style="color: red;"> b</span> and <span style="color: purple;">b'</span> rotated by an angle φ with respect to <span style="color: blue;">a </span>and <span style="color: lime;">a'</span>, the singlet state |Ψ-> leads to a value of S.</span></span><br />
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<span face="Helvetica, Verdana, Arial, sans-serif"><span style="line-height: 21px;">For some angles φ, S is out of bounds fixed by the Bell inequality</span></span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjCMeckWuCihAYYvFEsV_flbhuDk-IWqDp4U-gaxBzmHLDnLYfGEuZYR5ayRemLZUYuPNlNMVY8Hcse8Bls7KgVjJoTbDt8LniFLsOx4W_6jXCHup5I_0pCAm2WfSJS3lQkJdXQLXi6RmCM/s1600/bellcurve.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="277" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjCMeckWuCihAYYvFEsV_flbhuDk-IWqDp4U-gaxBzmHLDnLYfGEuZYR5ayRemLZUYuPNlNMVY8Hcse8Bls7KgVjJoTbDt8LniFLsOx4W_6jXCHup5I_0pCAm2WfSJS3lQkJdXQLXi6RmCM/s1600/bellcurve.png" width="400" /></a></div>
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<span face="Arial, Helvetica, sans-serif" style="color: #333333;">A measurement of polarization correlations for a proper relative orientation φ between the two sides should therefore probe the local realistic assumption behind the Bell inequality.</span><br />
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<a href="http://www.qolah.org/outreach/polanalyzer.png" style="clear: left; float: left; font-family: Helvetica, Verdana, Arial, sans-serif; margin-bottom: 1em; margin-right: 1em;"><img align="middle" alt="polarization analyzer" border="0" src="http://www.qolah.org/outreach/polanalyzer.png" style="width: 457.188px;" /></a><span style="color: #333333; font-size: 16px; line-height: normal;"><span face="Arial, Helvetica, sans-serif">The polarization of photons can be measured by a combination of beam splitters (BS), polarizing beam splitters (PBS) and half wave plates (HW), which divert an incoming photon onto one of four avalanche photodetectors. Those devices give a discrete arrival signal for incoming photons with a probability ("quantum efficiency") of about 50%.</span></span><br />
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The BB84 protocol allows us not only to test for eavesdropping, but also to guarantee that Alice and Bob can establish a secret key even if Eve has determined some of the bits in their shared binary sequence, using a technique called "privacy amplification". Imagine, for example, that Eve knows 10% of the key bits shared by Alice and Bob. Being aware of this, Alice and Bob could then publicly agree to add together (using modular arithmetic) each adjacent pair of bits to form a new sequence of half the length. Eve may also do this, but since she will need to know both bits in a pair in order to correctly determine their sum, she will find that she now shares a much lower fraction of the new bit sequence with Alice and Bob.</div>
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So much for the principle. In practice, pulses of single photons in a given quantum state are required for BB84. Recent progress using single atoms or semiconductor quantum dots can be used to generate single photons, however most practical QKD systems use weak laser pulses of different polarization states to send the bits that make up the key.<br />
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This method has an Achilles heel: the laser will sometimes generate pulses containing two or more photons, each of which will be in the same quantum state. As a result, Eve could in principle copy one of these photons and measure it, while leaving the other photons in the pulse undisturbed, thus determining part of the key while remaining undetected. Even worse, by blocking the single-photon pulses and allowing only the multi-photon pulses to travel through to Bob, Eve could determine the entire key. (This topic of hacking the signals is continued in the last section)</div>
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<span style="line-height: 1.4em;">Until true single-photon sources become available commercially, the most common defence is to strongly attenuate the laser to limit the rate of multi-photon pulses. However, this also means that many pulses contain no photons at all, reducing the rate at which the key can be transmitted. In 2003 a new trick to get round this problem was proposed by Hoi-Kwong Lo at the University of Toronto and Xiang-Bin Wang at the Quantum Computation and Information Project in Tokyo, based on earlier work by Won-Young Hwang at Northwestern University in the US.</span></div>
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Their idea was to intersperse the signal pulses randomly with some "decoy pulses" that are weaker on average and so very rarely contain a multi-photon pulse. If Eve attempts a pulse-splitting attack, she will therefore transmit a lower fraction of the decoy pulses to Bob than the signal pulses. Thus by monitoring the transmission of the decoy and signal pulses separately, Eve's attack can be detected. This means that stronger laser pulses may be used securely - for instance, last year at Toshiba we demonstrated a 100-fold increase in the rate that keys can be transmitted securely over a 25 km fibre. The decoy-pulse protocol has caused great excitement in the QKD community, with four independent groups having just reported experimental demonstrations of the technique.</div>
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Weak laser pulses are not the only way to carry out quantum cryptography. For example, QKD using a true single-photon source has recently been demonstrated at Stanford University, the CNRS in Orsay and Toshiba. Furthermore, in 1991 Artur Ekert, while a PhD student at the University of Oxford, described an alternative to the BB84 protocol that exploits another counterintuitive prediction of quantum mechanics: entanglement. Pairs of entangled photons have quantum states that are strongly correlated, such that measuring one photon affects the measurement of the other. If Alice and Bob each have one of the pair, they can therefore use their measurements to exchange information. This technique has been demonstrated by researchers at the University of Vienna, the Los Alamos National Laboratory and the University of Geneva, and was even used in 2004 to transfer money between Vienna City Hall and an Austrian bank. However, weak-laser QKD is the most mature approach, and the basis of the commercial QKD systems that are now coming on the market.</div>
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<strong><u><span style="font-size: large;">Practical Quantum Key Distribution - Polarization VS Phase Modulation</span></u></strong></h2>
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Information can be encoded in the quantum state of photons in several different ways. The first laboratory demonstration of QKD by Bennett and Brassard in 1989 over 30 cm of air used the polarization state of photons, which can be used to encode information via a polarization controller which can be decoded using a polarization analyzer.</div>
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<span face="arial, helvetica, sans-serif" style="background-color: white; font-size: 16px; line-height: 1.4em;">All polarized light can be described in terms of an electric field vector. This vector is a representation of the light’s electric field only (the magnetic field is not considered since it is proportional to and in phase with the electric field). </span></div>
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<span face="arial, helvetica, sans-serif" style="background-color: white; font-size: 16px; line-height: 1.4em;">For convention, this vector divided into two perpendicular components (x and y) where the third component (z) is simply the direction of wave propagation. When describing polarized light, two characteristics must be observed; First, the relative phase of the two components (x and y), and second, their relative amplitudes. These two characteristics determine the polarization of light. </span></div>
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<img height="205" src="http://hyperphysics.phy-astr.gsu.edu/hbase/phyopt/imgpho/polcls.gif" width="400" /></span><br /><span face="arial, helvetica, sans-serif" style="font-size: 16px; font-weight: normal;"><i>There are three main types of polarized light. They are linear, circular, and elliptical polarizations.</i></span><center style="color: #405679; font-family: arial, helvetica, sans-serif; font-size: 16px;">
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<br /><span face="arial, helvetica, sans-serif" style="font-size: 16px; font-weight: normal;"><u>Linear</u> polarization describes any light where the x-y components are in phase. The relative amplitude of these two components determines the direction of polarization (measured in radians from some reference point). Linear light is most easily obtained through use of a polarizer. The output light will always be linear, independent of input polarization (except in the case of absolute extinction where no output light is observed).<br /><br /><u>Circular</u> polarized light describes any light where the relative amplitudes are the same and there is a phase shift of exactly ninety degrees. Circular polarization is commonly described as either right-handed or left-handed. This can be visualized by imagining your thumb in the direction of propagation (z direction) and curling your fingers in the direction of the changing field. If this can be done with your right hand, it is obviously right-handed polarization, and left-handed polarization is found similarly. In reality, right-handed and left-handed circular polarizations are virtually the same. One would be concerned only in the case of mathematical convention.<br /><br />The third type of polarization, <u>Elliptical</u> polarization, describes any polarized light where relative phase and/or amplitude are not equal (excluding the circular and linear polarization cases). Elliptical light can be described as either right handed or left handed in a similar way as circular polarization. But in this case, it is often more important to describe whether the polarization is right handed or left handed. This ellipse can be described as seen in the figure below.</span><center style="color: #405679; font-family: arial, helvetica, sans-serif; font-size: 16px;">
<img alt="" height="149" src="http://www.photonics.byu.edu/images/poleell.GIF" style="height: 355px; width: 759px;" title="Courtesy:Agilent Technologies" width="320" /></center>
<br /><br /><span face="arial, helvetica, sans-serif" style="font-size: 16px; font-weight: normal;">Changes in both phase and relative amplitude can be monitored to characterize materials with variable birefringence. As the relative phase changes, the change can be described as an angle difference. The figure below illustrates a change in phase of the y component relative to the x component. If the relative amplitudes are the same, a full 360 degree shift may be viewed as the light changes through linear, elliptical, and circular polarizations. If the relative amplitudes are unequal, only linear and elliptical light will be observed. In either case, after a full rotation, the light will return to its original state.</span><br /><center style="color: #405679; font-family: arial, helvetica, sans-serif; font-size: 16px;">
<img alt="" src="http://www.photonics.byu.edu/images/pole%20change.GIF" style="height: 633px; width: 682px;" title="Courtesy:Agilent Technologies" /></center>
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The polarimetric testing process can be done manually with a set of polarizers and waveplates, such as with an apparatus below, where the looping of optical fiber can make elliptical polarization due to birefringence</span></span></span></h1>
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<span face="arial, helvetica, sans-serif">These mechanical polarization controllers have fiber loops curled up in paddles, which can be oriented with respect to each other by turning corresponding handles. </span><br />
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<span face="arial, helvetica, sans-serif">This is the way a QKD controller can can choose the measurement orientation of one of the detectors, to hit a condition to violate a Bell inequality quantum entangled photon .</span><br />
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<span face="arial, helvetica, sans-serif" style="background-color: white;">This can also be accomplished electronically with equipment such as the electro-optic polarization analyzer. The latter of which is much easier and more accurate but also is considerably more expensive. Both require the use of fiber-optic cables.</span></div>
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<span style="font-weight: normal;"><span style="font-size: small;">Circular polarized light is commonly launched into the fiber in order to reduce orientation dependence of the launching end. Also, we are able to easily monitor polarization changes in comparison to the launched circular light. The circular light is obtained through use of a polarizer and a quarter-wave plate preceding the fiber launch. At the detector end of the fiber, however, orientation relative to the analyzer is significant. The analyzer will read the x and y values corresponding to horizontal and vertical orientation of the input port, respectively.<br /><br />Throughout polarimetric testing it is important to promote mechanical and thermal stability as the fibers are commonly very sensitive to fluctuation. Even effects of the room’s ventilation, heating and air conditioning can be especially undesirable during testing. To increase thermal stability, a large heat sink such as a metal plate can be used. If the majority of the fiber is in thermal contact with this plate, any fluctuation will be relatively slow and uniform. The fiber may also be covered by another plate or box to protect from changes in the room’s air. Also, it is always wise to allow all equipment to warm up for at least thirty minutes before doing any type of test where high accuracy is desired.<br /><br />Commercial polarization analyzers can output a display as pictured below.</span></span><center>
<span style="font-weight: normal;"><span style="font-size: small;"><img alt="" src="http://www.photonics.byu.edu/images/display.GIF" style="height: 406px; width: 531px;" title="Courtesy:Agilent Technologies" /></span></span></center>
<span style="font-weight: normal;"><span style="font-size: small;"><br />Using the output display, the state of polarization can be monitored in both elliptical and spherical elliptical representations. The poincare sphere on the right is especially useful for polarization monitoring due to its ability to trace changes over time. The trace is marked by a red or blue line and will mark all polarization states recorded. The sphere monitors both relative amplitude and phase difference of the light’s polarization. Common locations are marked in the figure below.</span></span><center>
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The relative phase of the detected light is displayed as latitude on the sphere with circular polarizations at the poles. The relative amplitude (between x and y components of detected light) is displayed as longitude on the sphere. The four marked meridians correspond to horizontal, vertical, and 45 degree orientations of linear polarized light (at equator).<br /><br />The polarized light is most easily recorded as Stokes parameters. </span></span></h1>
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This is a vector < S0, s1, s2, s3> where:</span></span><center>
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<span style="font-weight: normal;"><span style="font-size: small;"><i>I </i>= total intensity<br /><i>p </i>= fractional degree of polarization (DOP)<br /><br />Using the Stokes Parameters, points on the sphere can be easily recorded and used in calculations. Commercial polarization analyzers can also do several calculations automatically such as angle change between given points</span></span></h1>
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<span face="sans-serif"><span style="background-color: white; line-height: 19.2px;">Polarization controllers can be operated without feedback, typically by manual adjustment or by electrical signals from a generator, or with automatic feedback in an electro-optic modulator that utilizes a electro-optic crystal such as Lithium Niobate. The latter allows for fast polarization manipulation and has been used in devices such as polarization scramblers for secure internet. Polarization scramblers usually vary the normalized Stokes vector of the polarization state over the entire Poincaré sphere. They are commercially available with speeds of 10 Mrad/s on the Poincaré sphere. Polarization controller technology is calable and great improvments have been made using electro-optic modulators, and minuture apparatus have been made which can encode polarization information on photons, such as with the PolaRite series of polarization controller devices:</span></span><br />
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The polarization controller technology is highly scalable and has been easily incorporated with microelectronics.</div>
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<span face="Helvetica, Verdana, Arial, sans-serif"><span style="line-height: 1.4em;">Scaling has </span><span style="line-height: 21px;">occurred</span><span style="line-height: 1.4em;"> too with the polarization analyzer technology. However the equipment is still quite large and expensive, and still too sensitive for use outside of the laboratory. Nevertheless, there has been great engineering advances over the last 20 years for it to become </span><span style="line-height: 21px;">feasible</span><span style="line-height: 1.4em;"> for use in QKD.</span></span><br />
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<span style="line-height: 1.4em;">Scaling and expense aside, the most important obstacle from a physics point of view is that transmitting photons along an optical fibre can randomize their polarization, which is disasterous for communication. A much better approach pioneered by Paul Townsend, formerly of BT Labs in the UK, is to alter the phase of the photon. </span><br />
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<span face="sans-serif" style="background-color: white; font-size: 13px; line-height: 19.05px;">For quantum phase estimation Suppose</span><span face="sans-serif" style="background-color: white; font-size: 13px; line-height: 19.05px;"> </span><img alt="U|u\big\rangle = e^{2\pi i\phi}|u\big\rangle" class="tex" src="http://ictwiki.iitk.ernet.in/wiki/images/math/e/f/0/ef0c86a23309b5d2b0de6ce55c63f5a9.png" style="border: none; font-family: sans-serif; font-size: 13px; line-height: 19.05px; margin: 0px; vertical-align: middle;" /><span face="sans-serif" style="background-color: white; font-size: 13px; line-height: 19.05px;">. The goal is to estimate</span><span face="sans-serif" style="background-color: white; font-size: 13px; line-height: 19.05px;"> </span><img alt="\phi\," class="tex" src="http://ictwiki.iitk.ernet.in/wiki/images/math/c/d/0/cd014731964c742c274df08d7cc238fb.png" style="border: none; font-family: sans-serif; font-size: 13px; line-height: 19.05px; margin: 0px; vertical-align: middle;" /><span face="sans-serif" style="background-color: white; font-size: 13px; line-height: 19.05px;">, but the trouble is, a overall global phase is irrelevant. Somehow it has to be turned into a relative phase. Here in comes the use of the idea of Eigenvalue Kickback. Suppose instead of</span><span face="sans-serif" style="background-color: white; font-size: 13px; line-height: 19.05px;"> </span><img alt="U\," class="tex" src="http://ictwiki.iitk.ernet.in/wiki/images/math/6/f/3/6f3d5ad4b0e22c80e2db450cf7238ea9.png" style="border: none; font-family: sans-serif; font-size: 13px; line-height: 19.05px; margin: 0px; vertical-align: middle;" /><span face="sans-serif" style="background-color: white; font-size: 13px; line-height: 19.05px;">, there is a two qubit operator</span><span face="sans-serif" style="background-color: white; font-size: 13px; line-height: 19.05px;"> </span><img alt="U_c\," class="tex" src="http://ictwiki.iitk.ernet.in/wiki/images/math/f/f/3/ff33523b8457bd9e9fcc025387066d75.png" style="border: none; font-family: sans-serif; font-size: 13px; line-height: 19.05px; margin: 0px; vertical-align: middle;" /><span face="sans-serif" style="background-color: white; font-size: 13px; line-height: 19.05px;"> </span><span face="sans-serif" style="background-color: white; font-size: 13px; line-height: 19.05px;">such that,</span><br />
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<a href="http://ictwiki.iitk.ernet.in/wiki/images/math/e/0/f/e0f924f49091385f5a33d7f347212d75.png" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img alt="\bullet U_c|0,\,u\big\rangle = |0,\,u\big\rangle" border="0" class="tex" src="http://ictwiki.iitk.ernet.in/wiki/images/math/e/0/f/e0f924f49091385f5a33d7f347212d75.png" style="border: none; vertical-align: middle;" /></a></div>
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<a href="http://ictwiki.iitk.ernet.in/wiki/images/math/1/8/8/1881f754758aa8394a7df1d66474fa75.png" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img alt="\bullet U_c|1,\,u\big\rangle = e^{2\pi i\phi}|1,\,u\big\rangle" border="0" class="tex" src="http://ictwiki.iitk.ernet.in/wiki/images/math/1/8/8/1881f754758aa8394a7df1d66474fa75.png" style="border: none; vertical-align: middle;" /></a></div>
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That is we have a controlled <img alt="U\," class="tex" src="http://ictwiki.iitk.ernet.in/wiki/images/math/6/f/3/6f3d5ad4b0e22c80e2db450cf7238ea9.png" style="border: none; margin: 0px; vertical-align: middle;" /> operation. Now if first qubit is <img alt="\frac{1}{\sqrt 2}(|0\big\rangle+|1\big\rangle)" class="tex" src="http://ictwiki.iitk.ernet.in/wiki/images/math/0/3/e/03e810d52b4f4327b89c263e990df84f.png" style="border: none; margin: 0px; vertical-align: middle;" />, then</div>
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<a href="http://ictwiki.iitk.ernet.in/wiki/images/math/e/c/1/ec1d2ca6f6d2e4dd0058218b719c5f15.png" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img alt="U_c\left(\frac{1}{\sqrt 2}(|0\big\rangle+|1\big\rangle)|u\big\rangle\right)=\frac{1}{\sqrt 2}(U_c|0,\,u\big\rangle+U_c|1,\,u\big\rangle)" border="0" class="tex" src="http://ictwiki.iitk.ernet.in/wiki/images/math/e/c/1/ec1d2ca6f6d2e4dd0058218b719c5f15.png" style="border: none; vertical-align: middle;" /></a></div>
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The "phase" has now landed up in the amplitude of the first photon and is measurable.</div>
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<span style="line-height: 1.4em;">In the experimental method, weak laser pulses are injected into an interferometer by Alice. By applying different voltages to a "phase modulator" in one arm of the interferometer, Alice can encode bits as a phase difference between the two emergent pulses sent to Bob - for example with 0° representing "0" and 180° representing "1". Bob then passes the pulses through another interferometer and determines which of his two detectors, corresponding to "0" and "1", they emerge at.</span><br />
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<tr><td align="center" class="caption">When using optical fibres for quantum key distribution, the bit values are usually encoded in the phases of individual photons by way of an interferometer. Photons generated by Alice can travel by one of two paths through her interferometer, and similarly through Bob's apparatus. As the path (green) through the short loop of Alice's interferometer and the long loop of Bob's is almost exactly the same length as the alternative route (purple) through Alice's long loop and Bob's short loop, the paths undergo optical interference. By applying a phase delay to each of the two paths, Alice and Bob can determine in tandem the probability that a photon will exit at either of Bob's detectors - corresponding to "0" and "1". For example, if Bob sets a phase delay of 0°, Alice can cause the photon to exit at "0" or "1" by applying phase delays to her modulator of 0° or 180°, respectively. To implement the BB84 protocol in this case, Alice applies one of four possible phase delays (&min;90°, 0°, 90°, 180°) to her modulator, in which a phase of 0° or 90° represents "0" and a phase of &min;90° or 180° represents "1". Meanwhile, Bob chooses a phase of either 0° or 90° with which to make his measurement. If the difference between Alice and Bob's phases is 0° or 180° then their choices are compatible, while if it is ±90° they are incompatible and Bob will measure a random bit value. Using a classical communication channel, Bob and Alice can then post-select their compatible choices to form a shared secret key.</td></tr>
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For this scheme to work, we must keep the relative lengths of the interfering paths in Alice and Bob's interferometers stable to a few tens of nanometres. However, temperature changes of just a fraction of a degree are enough to upset this balance. This was a roadblock in developing phase modulated QKD and allowed the polarization modulated QKD to gain important ground in research and development.</div>
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An ingenious solution to this problem was introduced by the Geneva group in 1997, which led to the first QKD system suitable for use outside the lab. The idea is to send the laser pulses on a round trip from Bob to Alice and then back to Bob so that any changes in the relative arm lengths are cancelled out. A QKD system based on this design is currently available for about €100,000 from the University of Geneva spin-out company id Quantique.</div>
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At the Toshiba lab in Cambridge, they had also developed an alternative compensation technique that allows pulses to be sent just one way, by sending an unmodulated reference pulse along with each signal pulse. These reference pulses are used as a feedback signal to a device that physically stretches the fibre in one of the two arms of the interferometer to compensate for any temperature-induced changes. In trials with the network operator Verizon, the one-way QKD system was continuously operated for over a month without requiring any manual adjustment.</div>
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We can assess the performance of QKD systems by the rate at which secure bits can be exchanged. The faster the secure-bit rate, the more frequently the key can be changed, thus inhibiting crypto-analysis. Typical secure-bit rates for complete QKD systems are in the range 10 - 50 kbit s - 1 for a 20 km fibre link. Although this may seem low compared with the rate data are transferred in optical communications (typically 1 - 40 Gbit s - 1), it is enough for up to 200 AES encryption keys (each of which comprises 256 bits) to be sent per second - sufficient for most cryptographic applications.</div>
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The secure-bit rate that can be achieved decreases with the length of the optical link due to the scattering of photons from the fibre. For this reason, the best performance is usually achieved using photons with a wavelength of 1.55 µm, at which standard optical fibre is most transparent. Even so, when the fibres get so long that the signal rate becomes comparable to the rate of false counts in Bob's photon detector, sending a secure key is no longer possible. For the standard indium gallium arsenide (InGaAs) semiconductor detectors used to detect 1.55 µm photons, this distance is currently about 120 km. Recently the Los Alamos group has used low-noise superconducting detectors to extend secure key distribution to fibres 150 km in length. Significantly, these distances are long enough for almost all the spans found in today's fibre networks.</div>
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Although the risk of cryptanalysis is mitigated by using QKD to frequently refresh the encryption key, it is not eliminated entirely. However, this can be achieved by encrypting the message using a "one-time pad", which requires a random key that contains the same number of bits as the message. Each bit of the message is then encrypted by adding it to the corresponding bit in the key using modular arithmetic. Provided that the key distribution is unconditionally secure, as it is using QKD, and that the key is never reused, the one-time pad is completely immune to attack. The downside is the length of the key that must be exchanged. QKD bit rates are already sufficient to allow unconditionally secret voice communication using the one-time pad. In the future, higher bit rates will allow this security to be extended to other forms of data.</div>
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Today's secure-bit rates are limited by how often the InGaAs detectors can detect a photon currently once every 100 ns. Silicon-based photon detectors can operate almost 1000 times faster, but they are only sensitive to shorter-wavelength photons. As the quality of InGaAs detectors improves over the next few years, we can expect their frequency to catch up with that of silicon, leading to QKD bit rates that are orders of magnitude higher. In the interim, there are encouraging results showing that non-linear crystals may be used to shift 1.55 µm photons to shorter wavelengths for which the faster silicon detectors may be used. Higher detection rates have also been demonstrated using superconducting nanowire detectors, and recent advances with detectors based on quantum dots are also encouraging.</div>
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<span style="font-size: large;"><u>Towards a quantum network </u></span></h2>
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One of the first real-life applications of QKD has been to secure fibre links between corporate sites in a city. Companies are increasingly using high-bandwidth optical connections between offices, data centres, server farms and disaster-recovery sites to obtain the speed and convenience of a local area network over a larger geographical area. In the early days of fibre deployment, immunity to "tapping" of sensitive data was often cited as a key advantage of fibre over copper cable. But in fact, eavesdropping on optical fibres can be accomplished by simply introducing a small bend in the fibre to extract a portion of the light; and, in the absence of quantum cryptography, it is almost impossible to detect performing such manipulations making fibre optic bugging a series threat to privacy and security.</div>
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To combat this, most companies develop so-called "link encryptors" that use the AES data encrption standard. However some companies, such as Toshiba, have already developed quantum link encryptor technology that can send data at 1 Gb/s between corporate sites, combining AES data encryption with secure key distribution using one-way QKD.</div>
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<tr><td align="center" class="caption">Toshiba's quantum-cryptography system consists of two boxes of optics and electronics, which sit at two sites connected by optical fibre and are designed to fit inside standard communications racks. All data fed into one unit are encrypted and transmitted via the fibre to the unit at the other site, where they are decrypted.<br />
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An important next step will be extending QKD from single point-to-point links into a "quantum network" for key distribution. Networks allow a company to connect multiple sites securely and to add new sites for an incremental cost. Moreover, they allow the range of QKD to be increased from the length of a single fibre link to any distance covered by the network, and safeguard against outages of individual links by automatically routing traffic around them.</div>
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In October 2003 BBN Technologies set up a primitive but pioneering QKD network in Cambridge, Massachusetts, linking their site with Harvard and Boston Universities. The firm showed that it was possible to direct the stream of single photons between different receiving units using an optical switch, and it also introduced the idea of "key relay" along a chain of trusted nodes. Here, each pair of adjacent nodes in the chain stores its own local key. A global key may then be sent from one end of the chain to the other, over any distance, by using the local keys and a one-time pad to encrypt each hop.</div>
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<span face="sans-serif" style="background-color: white; color: black; line-height: 19.1875px;">The rate of data transfer may be secure, but it has some limitations. Most obvious of which is the lack of speed, which follows from the no-cloning theorem, as QKD only can provide 1:1 connection. So the number of links will increase N(N-1)/2 as N represents the number of nodes. If a node wants to participate into the QKD network, it will cause some issues like destructing some quantum communication lines. </span>A more sophisticated system of QKD was launched by the European SECOQC consortium in 2008 to combat this, in a collaboration of academic and industrial QKD researchers, classical cryptographers and telecoms engineers. In Vienna, It has developed the first protocols required for routing, storage and management of keys within a meshed network. In the<span style="line-height: 1.4em;"> implementation of this quantum network it allows any two users at several sites across Vienna to establish a shared key.</span><span style="line-height: 1.4em;"> </span></div>
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SECOQC network architecture can by divided by two parts. Trusted private network and quantum network consisted with QBBs(Quantum Back Bone). Private network is conventional network with end-nodes and a QBB. QBB provides quantum channel communication between QBBs. QBB consists of a number of QKD devices that are connected with other QKD devices in 1-to-1 connection.<br />
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<tr><td align="center" class="caption">The 1-to-1 connection is a one-way weak pulse system (phase coding) designed by Toshiba (Tosh) Research Europe Ltd (TREL) is a fiber optic, decoy state system with phase encoding. It employs a decoy protocol using weak and `vacuum' pulses. This decoy protocol has been proven to be secure against all types of eavesdropping attacks. A single laser diode, operating with an intensity modulator, is used to produce signal and decoy pulses, so as to prevent attacks on any side channels that allow an eavesdropper to distinguish decoy pulses from signal ones.<br />
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The principle design of this one-way fiber optic QKD system uses two asymmetric Mach–Zehnder interferometers for encoding and decoding. Alice and Bob are linked by a quantum channel (optical fiber). The signal (an optical pulse with wavelength λ = 1.55 μm) is transmitted along the quantum channel at a repetition rate of about 7 MHz. The clock pulses (λ = 1.3 μm), which do not temporally overlap with the signal pulses, have a duration of 5 ns each and serve for synchronization purposes. An intensity modulator is used in order to produce signal and decoy pulses of different intensities at random times whereas vacuum decoy pulses are produced by omitting trigger pulses to the signal laser. The signal and decoy pulses are strongly attenuated to the single photon level, while a strong clock pulse is then multiplexed with them to provide synchronization. Bob's detectors are two single photon sensitive InGaAs avalanche photodiodes (APDs). The properties of the detectors are carefully adjusted in order to avoid so-called fake-state attacks and time-shift attacks. An active stabilization technique is used for continuous operation.<br />
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<span style="line-height: 1.4em; text-align: left;">Using the BB84 protocol, the sender (Alice) transmits photons to the recipient (Bob) in one of four different polarization states: horizontal (H), vertical (V), diagonal (D, 45°) and anti-diagonal (A, - 45°). For each photon she sends, Alice randomly selects one of these polarizations, with H or D representing the bit value "0" (red) and V or A representing "1" (blue), depending on the "basis" she chooses. To measure the photons, Bob is equipped with an analyser that can distinguish either between H and V (+) or between A and D (×). He randomly (and independently from Alice) chooses which analyser he will use to measure each photon. If Bob selects the analyser that is compatible with Alice's choice (top), he will determine the photon's polarization, and thus the bit value, with certainty. If, on the other hand, Bob measures with the "wrong" analyser (middle), he will obtain a random result.</span><br />
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It seems problematic that half of Bob's measurements result in a random bit value. However, Alice and Bob have a solution. After Bob's measurements have taken place, he reveals the sequence of analysers that he used. Alice then tells him which times he used the correct analyser, without revealing the bit that she sent. They can then discard all the measurements for which Bob used the wrong analyser, ensuring that they share the same bit sequence without any errors (in the absence of noise or imperfections).</div>
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This post-selection leaves an eavesdropper (Eve) at a disadvantage since she must guess which analyser to use to measure each photon (bottom). Inevitably Eve will sometimes select an analyser that is incompatible with Alice's choice of polarization, and thus may obtain a result that differs from the bit Alice sent. The key to the secrecy of quantum cryptography is that by making this measurement, Eve inevitably changes the quantum state of the photon. Therefore, when Bob receives the photon, he will sometimes determine an erroneous bit value even when he and Alice used compatible measurements. By examining a small sample of their bit sequence for errors, Alice and Bob can therefore determine whether an eavesdropper was present.</div>
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<i>Sketch of the optical layout of the TREL one-way weak coherent pulse QKD system (phase coding). The system represents a BB84 phase encoding protocol incorporating weak + vacuum decoy states. Atten.: attenuator, IM: intensity modulator, Pol: polarization controller, Φ: phase modulators, D1 and D2: avalanche photodiodes.</i><br />
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Secure QKD has been demonstrated over a number of fiber distances with this system. For a fiber distance of 20 km (4 dB loss), the SECOQC network running between Breitenfurterstrasse and Siemenstrasse obtained a secure bit rate of around 11 kbit s–1 over 24 h of continuous, autonomous operation. The secure bit rate reduces to an average of 5.7 kbit s–1 over a fiber length of 25 km (5 dB loss) over 24 h. This secure bit rate is almost six times higher than the SECOQC network specification of 1 kbit s–1 average secure bit rate over 25 km of fiber.<br />
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<tr><td align="left" style="padding: 10px 8px;">(a) The TREL quantum key distribution system. (b) Lower panel: secure bit rates for 24 h continuous operation for various fiber lengths: 20, 25 and 33 km. Upper panel: corresponding QBER for the various fiber lengths. (c) Secure bit rate as a function of fiber distance. Circles: experimental data derived from (b). Solid line: theoretical calculation optimized for a fiber length of 20 km.</td></tr>
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Meanwhile, id Quantique announced that it will install its "Vectis" link encryptor between the two centres of data-hosting company IX Europe in Zurich. In the US, MagiQ Technologies has recently developed its own encrypted link, targeted at government applications including the military, intelligence gathering and homeland security.</div>
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<span face="Helvetica, Verdana, Arial, sans-serif"><span style="line-height: 21px;"><i>(b) Secret key rate of system id Quantique1 over one day</i></span></span></div>
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<span face="sans-serif" style="color: black; line-height: 19.1875px;">From this, SECOQC can provide easier registration of new end-node in QKD network, and quick recovery from threatenings on quantum channel links. </span><span face="Arial, Helvetica, Verdana, sans-serif" style="color: black; line-height: 24.3px;">The nodes in question are situated in 19-inch racks. The photographs of the racks of nodes SIE, ERD, GUD, BRT and STP are shown below. </span></div>
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The principal design of the node module was carried out by the AIT Austrian Institute of Technology (formerly ARC ) in close collaboration with groups from University of Aarhus, Telecom ParisTech, University of Erlangen-Nuremberg, Bearing Point Infonova and Siemens Austria. The technical design and implementation of the module software was realized by a dedicated team from the AIT.</div>
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In what follows, we first of all give an account of the basic building blocks of the node module.<br />
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In the SECOQC approach, the main objectives of a node module are threefold:</div>
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to enable the functionality of all point-to-point QKD links connected to the node, to manage the key generated over these links, and on this basis, to ensure point-to-point ITS communication connectivity to all nodes in the network associated with the node by direct QKD links;</div>
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to determine a path from the node to any arbitrary destination node in the network along a sequence of nodes connected by direct QKD links; and</div>
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to ensure an end-to-end transport of secret key material along this path using the hop-by-hop transport mechanism.</div>
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These three distinct types of functionalities (or services) can be grouped in network layers: a quantum point-to-point (Q3P) layer, a quantum network layer and a quantum transport layer. A schematic representation of the node module design is given below.</div>
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<a href="https://www.blogger.com/blogger.g?blogID=5996268492156364201" name="nj313471fig18"><span style="color: black;"> <img alt="Figure 18" src="http://ej.iop.org/images/1367-2630/11/7/075001/Full/nj313471fig18.jpg" style="border: none;" /></span></a></div>
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<span class="sanserif" style="line-height: 25.92px;"><span face="Arial, Helvetica, Verdana, sans-serif"><i>Design of the node module.</i></span></span></div>
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<span face="Arial, Helvetica, sans-serif">The interface between the QKD device(s) and the node module on one side as well as between a pair of node modules is realized by the <i>quantum point-to-point protocol—Q3P</i>. </span></div>
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<span face="Arial, Helvetica, sans-serif">The node module sets up a Q3P connection to each node associated with it by a QKD link and thus initiates Q3P links.</span><span face="Arial, Helvetica, sans-serif" style="line-height: 1.35em;">A Q3P link realizes a context-dependent ITS communication channel. That is, depending on a header attached to the transmitted messages, it switches between different modes: (i) one-time pad encrypted and ITS authenticated communication, (ii) non-encrypted but ITS authenticated communication or (iii) non-encrypted and non-authenticated communication. To carry out this functionality each Q3P link maintains two functional entities: a</span><span face="Arial, Helvetica, sans-serif" style="line-height: 1.35em;"> </span><i style="font-family: Arial, Helvetica, sans-serif; line-height: 1.35em;">key store</i><span face="Arial, Helvetica, sans-serif" style="line-height: 1.35em;"> </span><span face="Arial, Helvetica, sans-serif" style="line-height: 1.35em;">(managing the key material), and a</span><span face="Arial, Helvetica, sans-serif" style="line-height: 1.35em;"> </span><i style="font-family: Arial, Helvetica, sans-serif; line-height: 1.35em;">crypto engine</i><span face="Arial, Helvetica, sans-serif" style="line-height: 1.35em;"> </span><span face="Arial, Helvetica, sans-serif" style="line-height: 1.35em;">(performing the crypto operations using key material from the key store), which are discussed below.</span></div>
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<span face="Arial, Helvetica, sans-serif">The Q3P protocol also maintains a communication line with the underlying QKD device corresponding to each Q3P link in the node. It takes care of transmitting transparently the QKD protocol messages between the peer QKD devices over the Q3P link, applying the security level, which is explicitly set in the QKD protocol communication calls. Additionally, Q3P accepts the key material that is pushed up by the QKD device and also passes over to it general node management commands.</span></div>
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The central element of this approach is the key store. The key store itself is organized in several levels:</div>
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<li><i style="font-family: Arial, Helvetica, Verdana, sans-serif;">Pickup store</i><span face="Arial, Helvetica, Verdana, sans-serif">: </span><span face="Arial, Helvetica, Verdana, sans-serif">More then one pair of QKD devices can be attached to a single Q3P link. Every QKD device is now associated with a pickup store to which it pushes the generated keys. There are no restrictions neither in size nor in time on the devices. However, finite size considerations related to privacy amplification indicate that reasonably large chunks of key materials are to be expected. </span><span face="Arial, Helvetica, Verdana, sans-serif">Note that the presence of key material in the pickup store has not yet been confirmed by the peer Q3P instance. Every chunk of key material has an unique identifier issued by the underlying QKD device. Using this identifier the peer key stores can perform a negotiation. A Q3P subprotocol is run, which ensures synchronous key presence on both sides. Once this protocol terminates successfully, the key material is moved to the common store.</span></li>
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<li style="font-family: Arial, Helvetica, Verdana, sans-serif;"><i>Common store</i>: There is only one single common store for the Q3P link, where all keys created by all QKD devices on the very same Q3P link are collected. Here, key boundaries as present in the pickup stores are disbanded and all chunks form a homogeneous mass of key bits. The common store is persistent and will be available after system reboot.</li>
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<li style="font-family: Arial, Helvetica, Verdana, sans-serif;"><i>In/out buffers</i>: As the communication over a Q3P link is bidirectional, pieces of key material have to be withdrawn from the common store to be dedicated for inbound or outbound communication. To prevent race conditions each key store participating on a Q3P link has one of two preselected roles—that of a <i>master</i> or of a <i>slave</i>. The master key store decides which concrete key material is to be withdrawn from the common store. The in/out buffers are crosswise interconnected. Once key material has been successfully used it is shredded and no longer available.</li>
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The key store architecture is schematically presented below:<br />
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<span face="Helvetica, Verdana, Arial, sans-serif"><span style="line-height: 22.4px;"><i>Architecture of the key store. Two QKD devices serving the same Q3P link are connected.</i></span></span></div>
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<span style="line-height: 1.4em;"><i>These QKD networks assume that the intermediate nodes are secure, which is realistic if the network is operated by a single service provider. </i></span></div>
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<span style="line-height: 1.4em;">In the future, however, we may be able to relax this condition using a device called a "quantum repeater". Quantum repeaters are based on the principle of quantum "teleportation", whereby a quantum state is transferred from one location to another, in principle over an arbitrary distance, using a pair of entangled particles, A and B. </span><span face="Arial, Helvetica, sans-serif" style="background-color: white; line-height: 1.4;">There is also a third photon, C, which is entangled to a quantum state but not to the other two photons. A quantum state may be represented by a group of atoms that shares a superposition between two ground states (in superposition, both ground states exist simultaneously, and there is a certain probability that the atoms are in one ground state or the other).</span></div>
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When physicists perform entanglement swapping by making a Bell state measurement on photons A and C, photon B also becomes immediately entangled to the quantum state, even though it has already traveled down the transmission channel.</div>
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The site where the sent photon becomes entangled with the quantum state is called a quantum repeater. Quantum repeaters, which occur throughout the transmission channel, can generate and store entanglement in order to boost the signal, with the aim of getting the entangled state to reach the other end. Entanglement can be stored by some sort of quantum memory device until, ultimately, the quantum state is “read” by being converted into another photon.</div>
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The key is that the quantum state should stay entangled with the sent photon for as long as possible, in order to maintain the quantum correlation throughout the transmission channel. Without a quantum memory, there is a small probability that quantum information can be transmitted over large distances, but the probability is exponentially dependent on the transmission length. With quantum memory, the transmission probability is only polynomially dependent on the length, greatly reducing the waiting time for a successful transmission.</div>
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<span style="line-height: 1.4em;"><span style="line-height: 1.4em;">Recent developments such as a semiconductor device for generating entangled photon pairs and the teleportation of quantum states between photons and atoms bring the quantum repeater closer to becoming a reality.</span><span style="line-height: 1.4em;"> </span></span></div>
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<span style="line-height: 1.4em;"><span face="Arial, Helvetica, sans-serif" style="background-color: white; line-height: normal;">(A) An atomic ensemble is confined in an optical trap formed by a focused laser beam. This beam is overlapped with counter-propagating “write” and “read” beams. The resulting Stokes and anti-Stokes photons are detected, serving as a useful probe for quantum memory storage. (B) An absorption image of the optically trapped atoms. Image Credit: Thorsten Strassel.</span></span></div>
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<span style="line-height: 1.4em;">Alternatives to using a fibre-optic network to send quantum keys over long distance may be to use free-space links to low-orbit communication satellites. In 2006 a collaboration between researchers at the universities of Vienna, Munich and Bristol implemented a free-space link over 144 km between Tenerife and La Palma to demonstrate free space, long distance QKD. Experiments with weather balloons have also been proposed which will eventually lead to more ambitious experiments with communications satellites. </span><br />
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Laser information transfer in general has also had ambitious project proposals, such as laser communication with Lunar Satellites which could see a quantum teleportation transfer to the moon one day, for scientific curiosity and as a tool for large baseline, high sensitivity interferometry to test aspects of General Relativity for example. .<br />
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However, if a global Quantum fiber network exists, satellite communications will most likely be classical encrypted signals, with the key transmitted over a secure quantum channel.<br />
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Therefore space-based quantum communication may be completely unnecessary but nevertheless an interesting demonstration of the technology. On the other hand quantum repeater technology, if scaled in a way comparable to modern Rubidium frequency standards and chip scale atomic clocks, could change this view entirely, leading to truly revolutionary quantum networks that could be even expanded by deep space probes carrying quantum memory devices.<br />
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<span style="line-height: 1.4em;"><u>Potential for Scaling </u></span></h2>
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<span face="Arial, sans-serif" style="background-color: white; line-height: 18px;">Large-scale integrated quantum photonic technologies will require on-chip integration of identical photon sources with reconfigurable waveguide circuits. Relatively complex quantum circuits have been demonstrated already, but few studies acknowledge the pressing need to integrate photon sources and waveguide circuits together on-chip. A key step towards such large-scale quantum technologies is the integration of just two individual photon sources within a waveguide circuit, and the demonstration of high-visibility quantum interference between them.</span></div>
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<span face="Arial, sans-serif" style="background-color: white; line-height: 18px;">In early 2014 </span><span face="Arial, sans-serif" style="background-color: white; line-height: 18px;">Scientists and engineers from an international collaboration led by the</span><span face="Arial, sans-serif" style="background-color: white; line-height: 18px;"> </span>University of Bristol<span face="Arial, sans-serif" style="background-color: white; line-height: 18px;"> have, for the first time, generated and manipulated single photons on a silicon chip</span><span face="Arial, sans-serif" style="background-color: white; line-height: 18px;">, a silicon-on-insulator device that combines two four-wave mixing sources in an interferometer with a reconfigurable phase shifter. </span></div>
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<span face="Arial, sans-serif" style="background-color: white; font-style: italic; line-height: normal;">Photonic quantum computer: two spontaneous photon-pair source are integrated within a tuneable Mach-Zehnder interferometer. The system is capable of generating and manipulating path-entangled two-photon states (credit: J. W. Silverstone et al./</span><em style="background-color: white; font-family: Arial, sans-serif; line-height: normal;">Nature Photonics</em><span face="Arial, sans-serif" style="background-color: white; font-style: italic; line-height: normal;">)</span><br />
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<span face="Arial, sans-serif" style="background-color: white; line-height: 18px;">They configured the device to create and manipulate two-colour (non-degenerate) or same-colour (degenerate) path-entangled or path-unentangled photon pairs. They observed up to 100.0 ± 0.4% visibility quantum interference on-chip, and up to 95 ± 4% off-chip. The attractiveness of the device removes the need for external photon sources, provides a path to increasing the complexity of quantum photonic circuits and is a first step towards fully integrated quantum technologies based on a silicon architecture. This is the clearest evidence yet that phase modulation is the best bet for scalable quantum optics for use in QKD.</span></div>
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<span style="font-size: large;"><u>Market for quantum cryptography </u></span></h2>
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From the first laboratory demonstrations over 30 cm of air to the latest fibre-based systems operating over 100 km, QKD has certainly come a long way in the last two decades. The technology has shrunk into compact units the size of typical network equipment and is fully automated. But despite the technical progress there are significant barriers to the adoption of new cryptographic technologies.</div>
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A particular problem for QKD is selling technology based on quantum mechanics to clients who often know little about physics and are used to traditional cryptography. Another hurdle is the lack of a security certification process for the equipment. Users need reassurance not only that QKD is theoretically sound, but also that it has been securely implemented by the vendors. It is encouraging that there are several initiatives under way to establish common security standards for QKD.</div>
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As the market for QKD develops, we can expect that the price of equipment will drop significantly. Within 10 years we may see QKD used not only in corporate and government networks, but also in networks serving home users. Optical fibres are already used to deliver television, phone and Internet services to domestic users in several countries. Although current QKD systems are too expensive for such applications, they may become viable if miniaturization to microchip-scale and mass-production lead to the expected price reductions. The days when the products of the quantum-information industry serve every household may not be too distant.</div>
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<span style="font-size: large;"><u>Portable Quantum Cryptography Devices</u></span></h2>
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Researchers at labs in Los Alamos National Laboratory, America and Bristol, UK have both been developing small scale quantum key distribution devices which utilize polarization controllers. This is a significant step and is one that warrants attention as it is the first step towards developing devices which could carry QKD technology. </div>
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<span face="Arial, Verdana, Helvetica, sans-serif" style="background-color: white; line-height: 17.92px;">The new Los Alamos hardware </span><span face="Arial, Verdana, Helvetica, sans-serif" style="background-color: white; line-height: 17.92px;">employs a laser together with variable-direction polarization components to establish a secret key between sender and receiver. The key could be used to encrypt (and later decrypt) a message sent by any means—over the Internet, phone, satellite, or even carrier pigeon. That is, only the key generation requires a quantum channel—photons, polarizations, and fiber optics. Once the key is established, the encrypted information can be sent by any available means, including wireless.</span></div>
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a new technology they call QKarD. (The capitalized letters stand for quantum key distribution.) QKarD is a small, handheld smart card that can supply quantum keys for a variety of uses. When the card is docked in its charging station with a fiber-optic network, it automatically establishes the next thousand or so keys it will need and stores them in secure memory. The card can then be undocked and carried around by its owner.</div>
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Because quantum keys must be established between a sender and receiver, and the QKarD doesn't know who the next receiver will be while it is docked, an external server is needed to manage all transmissions. This server would reside with a trusted private or government agency. A docked QKarD establishes keys between itself and this agency.</div>
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When the QKarD owner wants to send a secure transmission—for example, to transmit her credit card number to an online bookstore—she connects the QKarD to a computer or other mobile device. Her credit card number is encrypted with the next available quantum key on the QKarD, and the transmission is sent to the bookstore. The bookstore notifies the trusted agency that the transmission occurred but does not share the encrypted credit card number with that agency.</div>
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The bookstore uses QKarD technology just as the consumer does, so it has its own quantum keys already established with the same trusted agency. At the time of purchase, the agency tells the online bookstore which bits to flip (from 0 to 1 or vice versa) to transform the bookstore's next available quantum key into the same quantum key from the buyer's QKarD. In this way, a common quantum key is established between the buyer and seller with the help of the trusted agency, but without the buyer needing a fiber-optic connection at the time of the purchase.<br />
This QKarD system is completely mobile, apart from the need to be docked from time to time to acquire quantum keys while charging. Therefore it could serve all wireless transmissions—laptop computing, cell phone calls, e-commerce, and so forth. For example, QKarD technology could be integrated into a future generation of smart phones. Every transmission from every app would be secure.</div>
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Scrambled Bits</span></span></h3>
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The Los Alamos team has also created a second prototype system for enhanced security and privacy. Known as quantum enabled security (QES), it is both a device and a communications protocol. It uses QKarD keys to hide data transmissions from potential eavesdroppers.</div>
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Some networks that provide fiber-optic lines to homes, for example, employ 32 wavelengths simultaneously to 32 homes. The receiving electronics system in each home is programmed to pay attention to just one particular wavelength and ignore the other 31. So everyone's data also goes into 31 other homes. But the QES system uses a secure quantum key to obscure every message in both wavelength and time. Messages are scrambled with different bits hopping among different wavelengths at different moments. Only the authorized receiver's hardware, which has the key, knows how to unscramble the hopping and pay attention to only the right wavelengths at the right moments, to pick up every 0 and 1. This is known as physical-layer security—the most robust kind—because one needs to physically select the right bits to acquire the hidden, encrypted message. While other security schemes allow eavesdroppers to record encrypted messages, eavesdroppers on a QES system can't even identify which bits contain the encrypted message.</div>
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The QES technology is limited by the distance a single photon can travel through fiber-optic cables without its polarization degrading. That distance is at least 140 kilometers (87 miles), meaning that hardware needs to be placed at stations no more than 140 kilometers apart. (It is possible to break a 280-kilometer transmission, for example, into two 140-kilometer transmissions that each have their own key.) Until such hardware is set up throughout the country or the world, QES is ideal for more localized security, such as between buildings on the same campus, industrial site, neighborhood, or metropolitan center. It is also well suited for securing information within isolated entities, such as U.S. embassies abroad or national security facilities like Los Alamos National Laboratory.</div>
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<span face="Helvetica, Verdana, Arial, sans-serif" style="line-height: 21px;">Also, as discussed earlier, the polarization control technique is limited to small scale networks due to the fact that randomization occurs to the polarization state over large distances. For large scale, phase modulation is necessary along with quantum repeater technology. In this way, using existing architecture on telecommunications, more secure financial transactions could arise; replacing debit and credit cards with mobile phone application software with access to QKD servers, at a bank or phone service provider.</span></div>
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<span face="Arial, Helvetica, sans-serif">Photon pairs with a strong correlation in time can not only be prepared from cascade decays in atoms, such as in single photon laser sources, but also in a nonlinear optical process called spontaneous parametric down conversion, SPDC. </span></div>
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<span face="Arial, Helvetica, sans-serif">In appropriate materials, photons from a strong light field can be "broken up" into pairs of photons of lower energy, as long as the total energy and, depending on the boundary conditions, the total momentum is conserved.</span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhp4yReqVTIB5dk0wvRdNnJdBFvOoCBwzNHhN471iik67RSx-khQ3rPm3b5Uxtix9O4Fj40crdT8HiC1wlFdsbrI419gfxC_9guZi0h4we_YCSnGdf6ousCs_pzyi4Oew1NZr3b9_jIafYr/s1600/spdc.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="196" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhp4yReqVTIB5dk0wvRdNnJdBFvOoCBwzNHhN471iik67RSx-khQ3rPm3b5Uxtix9O4Fj40crdT8HiC1wlFdsbrI419gfxC_9guZi0h4we_YCSnGdf6ousCs_pzyi4Oew1NZr3b9_jIafYr/s1600/spdc.jpg" width="400" /></a></div>
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<span face="Arial, Helvetica, sans-serif">Entangled pair sources have seen a dramatic improvement in the last few years; light is now generated into optical fibers, and can be conveniently transported. They also help to block unwanted light wavelengths, creating a coherent entangled photon source:</span></div>
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Addition of compensator crystals to the polarized photon source has the overall effect of increasing the time correlation between the 2 entangled photons by several orders of magnetude.</div>
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This is a technique employed in polarization microscopy, which produces highly accurate analytical instruments that can be employed to determine the <b>relative retardation</b> (often symbolized by the Greek letter <b>Γ</b>) or optical path difference between the orthogonal wavefronts (termed <b>ordinary</b> and <b>extraordinary</b>) that are introduced into the optical system by specimen birefringence, in this case the birefringence of the BBO crystal itself. Normally this is compensated by placing the crystal in an oven, however this is energy intensive, bulky and damages the crystal in the long run. </div>
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The terms relative retardation, used extensively in polarized light microscopy, and optical path difference (<b>Δ</b> or <b>OPD</b>), are both formally defined as the relative phase shift between the orthogonal wavefronts, expressed in nanometers, according to the equation:</div>
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<b>Δ = Γ = t • (n<sub style="font-size: 0.75em;">e</sub> - n<sub style="font-size: 0.75em;">o</sub>)</b></center>
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where <b>t</b> refers to the thickness of the specimen (the physical distance traversed by light waves through the specimen), <b>n(e)</b> is the refractive index experienced by the extraordinary wavefront, and <b>n(o)</b> is the refractive index experienced by the ordinary wavefront.<br />
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The terms (refractive index and thickness) on the right side of the equation are collective referred to as the specimen birefringence. From this relationship, it is obvious that specimens having differing thickness and refractive index gradients can display identical optical path differences or relative retardations. Furthermore, if either the birefringence or the thickness of a specimen is known, the other parameter can be easily determined.</div>
<span face="Arial, Helvetica, sans-serif" style="background-color: white; font-size: 14px; line-height: normal;">Compensators are usually composed of optically anisotropic quartz, mica, and gypsum minerals ground to a precise thickness and mounted between two optical windows having flat (plane) faces, which are designed to introduce a fixed amount of retardation between the orthogonal wavefronts passing through the crystal. Needless to say, these compensators are expensive, as are quartz waveplates in general. More recently, several manufacturers have shifted to the application of a highly aligned and stretched linear organic polymer to produce anisotropic retardation plates to combat costs.</span><br />
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<span face="Helvetica, Verdana, Arial, sans-serif" style="line-height: 21px;"><br /><br /><br />MuonRay Enterprises Quantum Entanglement Experiments - Opensource Quantum Random Number Generation for Cryptography<br /><br /></span>
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<span face="Helvetica, Verdana, Arial, sans-serif" style="line-height: 21px;">Interested by these developments, I took it upon myself to see if I could construct a portable quantum optics device myself, using a non-linear crystal of Beta-Barium Borate (BBO) as a parametric down-converter for 405nm laser light. </span><br />
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<span style="background-color: white;"><span face="Verdana, Geneva, Arial, Helvetica, sans-serif"><span style="line-height: 18.975px;"><i>Crystal of Beta-Barium Borate (BBO) performing Spontaneous Parametric Down Conversion (SPDC) of 405nm laser light. This process can form random entangled photon pairs.</i></span></span></span><br />
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Below is a video of such an experiment performed in 2013:</div>
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<span face="Arial, Verdana, Helvetica, sans-serif"><span style="line-height: 21px;">The UV laser diode is </span></span><span face="Helvetica, Verdana, Arial, sans-serif" style="line-height: 21px;">connected</span><span face="Arial, Verdana, Helvetica, sans-serif"><span face="Helvetica, Verdana, Arial, sans-serif"><span style="line-height: 21px;"> to the BBO crystal in an assembly which can rotate the crystal about its crystal axis. The </span></span><span style="line-height: 21px;">assembly</span><span face="Helvetica, Verdana, Arial, sans-serif"><span style="line-height: 21px;"> is self contained and requires no optical bench.</span></span></span><br />
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<span face="Arial, Verdana, Helvetica, sans-serif"><span style="line-height: 21px;">Both pair photons are generated at the same wavelength (810nm) or at different wavelengths. </span></span><br />
<span face="Arial, Verdana, Helvetica, sans-serif"><span style="line-height: 21px;"><br /></span></span><span face="Arial, Verdana, Helvetica, sans-serif"><span style="line-height: 21px;">The use of a beamspliiter creates a coherent source of entangled photons. Hence we can consider two different approaches to </span></span><br />
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<span face="Arial, Verdana, Helvetica, sans-serif"><span style="line-height: 21px;">(a) Detection by beamsplitter approach employs the particle nature of photons.</span></span><br />
<span face="Arial, Verdana, Helvetica, sans-serif"><span style="line-height: 21px;">In this design, detection </span><span style="line-height: 21px;">obtained this way will always be somewhat biased due to the inevitable imbalance in photon detection rates between the two detectors. photons here are from an incoherent source, i.e. starlight or lightbulb.</span></span><br />
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<span face="Arial, Verdana, Helvetica, sans-serif"><span style="line-height: 21px;">(b)Detection by wavefunction collapse employs wave-like nature of photons.</span></span><br />
<span face="Arial, Verdana, Helvetica, sans-serif"><span style="line-height: 21px;">In this design, the photon detection rate imbalance is avoided by having a coherent, long-lasting time correlated entangled photon which will collapse into a time window defined by a measurement (provided that each measurement time window is much smaller than the photon coherence time itself).</span></span><br />
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Using entangled photons from a 405nm laser diode pump as the source with a coherence time of 1ms, while the detector is gated at 1GHz with each detection window 1000 times smaller than the coherence time. Such a large disparity in time ensures an equal detection probability between any two adjacent detection gates. By assigning a bit value ‘0’ or ‘1’ according to a detection event at an even or odd clock cycle, a bias-free random number is readily obtainable.</div>
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<b>Figure 2. </b>Photon count rate as a function of incident light intensity recorded for a self-differencing indium gallium arsenide single-photon APD. A maximum photon count rate of 497MHz is measured, which is very close to the value expected from a theoretical calculation (black line) assuming zero detector dead time.</div>
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To realize this scheme, it is crucial for the detector to have certain characteristics. Ideally, the detector should be operated in gated mode in order to allow unambiguous bit-value assignments. Moreover, to achieve high bit rates that are free of bias, the detector must be able to handle high photon rates and possess a negligible counting dead time.</div>
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Semiconductor avalanche photodiodes (APDs) are well suited to this task. Operated under a self-differencing mode<a href="http://spie.org/x35516.xml#B4" style="color: #a00b10; font-family: Helvetica, Arial, sans-serif;"><span style="bottom: 0.7ex; height: 13px; position: relative; vertical-align: baseline;">4</span></a> that was originally devised by us for megabit-per-second (Mb/s) secure key-rate quantum key distribution,<a href="http://spie.org/x35516.xml#B1" style="color: #a00b10; font-family: Helvetica, Arial, sans-serif;"><span style="bottom: 0.7ex; height: 13px; position: relative; vertical-align: baseline;">1,2</span></a>an analog telecommunication APD can be converted into a high-speed single-photon detector. Using this system, we achieved a record photon count rate of ~500MHz and an ultra-short dead time of less than 2ns (see Figure <a href="http://spie.org/x35516.xml#fig2" style="color: #a00b10; font-family: Helvetica, Arial, sans-serif;">2</a>).<a href="http://spie.org/x35516.xml#B5" style="color: #a00b10; font-family: Helvetica, Arial, sans-serif;"><span style="bottom: 0.7ex; height: 13px; position: relative; vertical-align: baseline;">5</span></a></div>
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<b>Figure 3. </b>Byte correlation pattern of 500m bits generated by our 52Mb/s quantum RNG.</div>
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Incorporating a self-differencing APD, we have initially realized a quantum RNG with a random bit stream of 4Mb/s.<a href="http://spie.org/x35516.xml#B3" style="color: #a00b10; font-family: Helvetica, Arial, sans-serif;"><span style="bottom: 0.7ex; height: 13px; position: relative; vertical-align: baseline;">3</span></a> Importantly, the quantum randomness has survived in this physical realization, and the random number outputs are intrinsically free of bias and do not require mathematical post-processing to pass random number statistical tests.</div>
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It is the first time that any quantum RNG—and perhaps any physical RNG—has not required post-processing to pass stringent randomness tests. Furthermore, by using finer photon timing, the random bit rate can be increased by over an order of magnitude to 52Mb/s with no degradation in randomness.<a href="http://spie.org/x35516.xml#B6" style="color: #a00b10; font-family: Helvetica, Arial, sans-serif;"><span style="bottom: 0.7ex; height: 13px; position: relative; vertical-align: baseline;">6</span></a> Figure <a href="http://spie.org/x35516.xml#fig3" style="color: #a00b10; font-family: Helvetica, Arial, sans-serif;">3</a> shows a visualization of the random output from our 52Mb/s RNG.</div>
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Despite the state-of-the-art performance, the bit rate must be improved to serve demanding applications, such as high bit-rate quantum key distribution.<a href="http://spie.org/x35516.xml#B1" style="color: #a00b10; font-family: Helvetica, Arial, sans-serif;"><span style="bottom: 0.7ex; height: 13px; position: relative; vertical-align: baseline;">1,2</span></a> Presently, the random bit rate is limited by the photon recording electronics, which can manage photons at only 5 million per second.</div>
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In future work, we aim to design electronics to cope with a photon rate of 1 billion per second and with such technology available, we expect to see the bit rate surpassing 100Mb/s. Using finer timing, a bit rate of multi-gigabits per second is in sight.</div>
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<span face="Arial, Verdana, Helvetica, sans-serif"><span style="line-height: 21px;">Silicon Avalanche Photodiodes (APDs), of the kind used in military laser rangefinders and hyperspectral CCDs, are used to detect the 810 nm photons.</span></span><br />
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<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhs5MfOeja3FBsmpp_qLwU6f12JznJhR_-b180XS3EBJDpo9NsYxi8ulvxF3Vwj65a3uyRiIUBlCmaQ1SdTyUm_Ej9GYxJq_LkOeqBrkBIgLIjia5-9QTWKQ1N1GgDvuJ2LxRL24NmcNiM9/s1600/silicon+avalanche+photodiodes.jpg" /></div>
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<span face="Arial, Verdana, Helvetica, sans-serif" style="line-height: 21px;">For the mode where 2 photons of different frequencies are generated, 680nm for Alice and 1550nm for Bob, Silicon Single Photon Avalanche Photodiodes (Si-SPADs) have perfect properties for the 680nm photons at Alice's side. </span><br />
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<span face="Arial, Verdana, Helvetica, sans-serif"><span style="line-height: 21px;">Bob's</span></span><span face="Arial, Verdana, Helvetica, sans-serif" style="line-height: 21px;"> telecom 1550nm-photon has low transmission losses in optical fibers. Bob's photon is detected by InGaAs-APDs that need to be gated as usual. </span><br />
<span face="Arial, Verdana, Helvetica, sans-serif" style="line-height: 21px;">Therefore an optical trigger pulse co-propagates with each signal photon to open the detector for few nanoseconds. </span><br />
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<span face="Helvetica, Verdana, Arial, sans-serif"><span style="line-height: 21px;">The pump 405nm laser diode is itself connected to a circuit that can send signals via the </span></span><span face="Helvetica, Verdana, Arial, sans-serif" style="line-height: 21px;">Phillips</span><span face="Arial, Verdana, Helvetica, sans-serif"><span face="Helvetica, Verdana, Arial, sans-serif"><span style="line-height: 21px;"> RC-5 protocol, borrowing from its </span></span><span style="line-height: 21px;">success</span><span face="Helvetica, Verdana, Arial, sans-serif"><span style="line-height: 21px;"> and ubiquity in IR sensor and controller technology. </span></span></span><br />
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The information sheet provided below is the most complete and accurate information on the number allocations and RC-5 commands available at this time. It is from a printed document from Philips dated December 1992.</div>
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<a class="image" href="http://en.wikipedia.org/wiki/File:Rc-5_protocol_details.jpg" style="background-image: none;"><img alt="Rc-5 protocol details.jpg" height="640" src="http://upload.wikimedia.org/wikipedia/commons/c/c7/Rc-5_protocol_details.jpg" style="border: none; margin: 0px; vertical-align: middle;" width="529" /></a></div>
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<span face="Helvetica, Verdana, Arial, sans-serif"><span style="line-height: 21px;">The RC-5 code from Philips is possibly the most used protocol by hobbyists, probably because of the wide availability of cheap remote controls. </span></span><br />
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<span face="Helvetica, Verdana, Arial, sans-serif"><span style="line-height: 21px;">C5 is stream of 14 equal length bits of exactly 1.778ms per bit time. </span></span><br />
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<span face="Helvetica, Verdana, Arial, sans-serif"><span style="line-height: 21px;">A pulse during the first half of the clock time represents 0, a pulse in the second half of clock time represents 1. This scheme is called Manchester coding</span></span><br />
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<img height="189" src="http://upload.wikimedia.org/wikipedia/commons/thumb/9/90/Manchester_encoding_both_conventions.svg/650px-Manchester_encoding_both_conventions.svg.png" width="400" /><br />
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<span face="Helvetica, Verdana, Arial, sans-serif"><span style="line-height: 21px;">This form of coding, converts the original data into the Manchester value by means of a clock XOR gate.</span></span><br />
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<span face="Helvetica, Verdana, Arial, sans-serif"><span style="line-height: 21px;"> original data = clock XOR = Manchester value</span></span><br />
<span face="Helvetica, Verdana, Arial, sans-serif"><span style="line-height: 21px;"> 0 0 0</span></span><br />
<span face="Helvetica, Verdana, Arial, sans-serif"><span style="line-height: 21px;"> 0 1 1</span></span><br />
<span face="Helvetica, Verdana, Arial, sans-serif"><span style="line-height: 21px;"> 1 0 1</span></span><br />
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<span face="Helvetica, Verdana, Arial, sans-serif"><span style="line-height: 21px;"> 1 1 0</span></span><br />
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<span face="Helvetica, Verdana, Arial, sans-serif" style="line-height: 21px;">Manchester encoding is a special case of binary phase-shift keying (BPSK), where the data controls the phase of a square wave carrier whose frequency is the data rate. Such a signal is easy to generate.</span><br />
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<span face="Helvetica, Verdana, Arial, sans-serif"><span style="line-height: 21px;">The bits are now represented by two phases, on the real axis, at 0° and 180°.</span></span><br />
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<img src="http://upload.wikimedia.org/wikipedia/commons/thumb/4/41/BPSK_Gray_Coded.svg/200px-BPSK_Gray_Coded.svg.png" /><br />
<span face="Helvetica, Verdana, Arial, sans-serif"><span style="line-height: 21px;">This is the classical 2D constellation interpretation of the bits, the qubits will be on opposite points on the 3D Bloch sphere.</span></span><br />
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<span face="Helvetica, Verdana, Arial, sans-serif" style="line-height: 21px;">The first two bit times, S1 and S2 are start bits, followed by a toggle bit, T. The toggle bit inverses each time a button is pressed so the receiver can tell the difference between a hold and a repeated press. The next 5 bits are the address (0b11110=0x1E), followed by the command (0b000001=0×01, 0b000010=0×02). A backwards compatible extension to RC5 uses the second start bit as command bit 7.</span><br />
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<span face="Helvetica, Verdana, Arial, sans-serif"><span style="line-height: 21px;">Phillips itself has since moved to RC-6, probably in response to semiconductor companies in china making clones of its CMOS VCO chip technology.</span></span><br />
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<span face="Helvetica, Verdana, Arial, sans-serif"><span style="line-height: 21px;">Therefore, using the various protocols used in infrared communication, there is already an electronics infrastructure in existence which can be utilized to test quantum key encryption in free space with relative low cost.</span></span><br />
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<span face="Verdana, Geneva, Arial, Helvetica, sans-serif" style="background-color: white; line-height: 18.975px;">The </span><span face="Verdana, Geneva, Arial, Helvetica, sans-serif" style="background-color: white; line-height: 18.975px; margin: 0px; padding: 0px;">principal idea</span><span face="Verdana, Geneva, Arial, Helvetica, sans-serif" style="background-color: white; line-height: 18.975px;"> of this setup is to use the unique quantum mechanical property of “entanglement” in order to transfer the correlated measurements into a secret key. </span><br />
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<span face="Verdana, Geneva, Arial, Helvetica, sans-serif" style="background-color: white; line-height: 18.975px;">Using a combination of our knowledge of generating entangled photons by SPDC and the </span><span face="Helvetica, Verdana, Arial, sans-serif" style="line-height: 21px;">Phillips</span><span face="Arial, Verdana, Helvetica, sans-serif"><span face="Helvetica, Verdana, Arial, sans-serif"><span style="line-height: 21px;"> RC-5 protocol, prototypes of a portable quantum entanglement device were developed.</span></span></span><br />
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<span face="Verdana, Geneva, Arial, Helvetica, sans-serif" style="background-color: white;"><span style="line-height: 18.975px;">The first prototype is a system that interfaces with a digital to analog system to send a key from a computer to the entanglement source which then sends the pairs of entangled photons which can be received via a parabolic dish which focuses the light to an infrared sensor.</span></span><br />
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<span face="Verdana, Geneva, Arial, Helvetica, sans-serif" style="background-color: white;"><span style="line-height: 18.975px;">The second prototype uses a fiber optic link between the emitter and receiver in a more compact circuit.</span></span><br />
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<span face="Verdana, Geneva, Arial, Helvetica, sans-serif" style="background-color: white;"><span style="line-height: 18.975px;">The third prototype uses a highly compact transceiver system that can send and receive quantum keys. The receiver system is also a new flexible and transparent fiber optic relay that can cover a wide area and is flat so that it can be hidden on the side of the device until deployed. such a system is particularly useful for making the technology portable </span></span><span face="Verdana, Geneva, Arial, Helvetica, sans-serif" style="background-color: white;"><span style="line-height: 18.975px;">via a single unit which could be designed around the perimeter of a functioning device, such as a handheld device (phone, tablet computer, ect).</span></span><br />
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<span style="background-color: white;"><span face="Verdana, Geneva, Arial, Helvetica, sans-serif"><span style="line-height: 18.975px;">Using existing detector infrastructure, a passive system performing measurements can be implemented, where all photons find their way towards their detectors without the need to control any of their properties actively. As a result, correlated measurements are generated at Alice and Bob without any input for choice of basis or bit value for individual qubits. </span></span></span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgDkhevP_tyufcK5xpzx7jD1M_WfMnMu5ntPQjEaaQtduFGaFrqSmd_Yf3dS9rWhKKjma0neh3W9S9G1T1O3-eotoWXyl-2XsQiLIusfRb72YugEJuEwW3hTP_CLaNoRZRQKYjcLQPfIR7C/s1600/enta_asymm_src_setup.jpg" style="margin-left: 1em; margin-right: 1em;"><span style="color: black;"><img border="0" height="217" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgDkhevP_tyufcK5xpzx7jD1M_WfMnMu5ntPQjEaaQtduFGaFrqSmd_Yf3dS9rWhKKjma0neh3W9S9G1T1O3-eotoWXyl-2XsQiLIusfRb72YugEJuEwW3hTP_CLaNoRZRQKYjcLQPfIR7C/s1600/enta_asymm_src_setup.jpg" width="400" /></span></a></div>
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<i><span face="Verdana, Geneva, Arial, Helvetica, sans-serif" style="background-color: white; line-height: 18.975px; margin: 0px; padding: 0px;">QKD-system using entangled photons measured at Alice and Bob. The correlated measurements from single-photon detectors are further processed and transferred to a symmetric, secure key by the QKD software.</span><span face="Verdana, Geneva, Arial, Helvetica, sans-serif" style="background-color: white; line-height: 18.975px;"> Alice, the server, uses a low-intensity, short range 680nm channel, where as Bob, the client, uses a relatively more robust telecom 1550nm channel. </span></i><br />
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<span face="Verdana, Geneva, Arial, Helvetica, sans-serif" style="background-color: white; line-height: 18.975px;">For long-distance quantum fiber-communication systems it is essential to have a high flux of photon pairs generated by spontaneous parametric down conversion i</span><span style="background-color: white;"><span face="Verdana, Geneva, Arial, Helvetica, sans-serif"><span style="line-height: 18.975px;">n the orthogonally oriented crystal geometry. </span></span></span><br />
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<span style="background-color: white;"><span face="Verdana, Geneva, Arial, Helvetica, sans-serif"><span style="line-height: 18.975px;">Even the best non-linear crystals have limits on their efficiency at generating pairs of entangled photons. To increase the conversion efficiency, a periodically poled nonlinear crystal with a high non-linear coefficient, such as BBO, can be used. </span></span></span><br />
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<span face="sans-serif"><span style="background-color: white; line-height: 19.2px;">Periodic poling is a formation of layers with alternate orientation in a birefringent nonlinear crystal. The crystal domains are regularly spaced, with period in a multiple of the desired wavelength of operation. The structure is desired to achieve quasi-phase-matching (QPM) in the material.</span></span><br />
<span face="sans-serif"><span style="background-color: white; line-height: 19.2px;">Tests have shown that, with periodic poling, the crystals are up to 20 times more efficient at second-harmonic generation than crystals of the same material without periodic structure.</span></span><br />
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<span face="Verdana, Geneva, Arial, Helvetica, sans-serif" style="background-color: white; line-height: 18.975px;">A compact entangled photon source can be pumped by a 405-nm-laser and its polarization is rotated to 45° for equal crystal excitation. The nonlinear ppBBO crystal is quasi-phase matched for all three wavelengths involved, 405nm, 680nm, 1550nm.</span><span face="Verdana, Geneva, Arial, Helvetica, sans-serif" style="background-color: white; line-height: 18.975px;"> </span><br />
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<i style="background-color: white; font-family: Verdana, Geneva, Arial, Helvetica, sans-serif; line-height: 18.975px; margin: 0px; padding: 0px;">Schematic of the source of entangled photons. Within the nonlinear, periodically poled BBO crystal, single 405nm pump photons are converted to two photons at 810nm in the H and V modes, polarization entanglement is then generated.</i><span face="Verdana, Geneva, Arial, Helvetica, sans-serif" style="background-color: white; line-height: 18.975px;"> </span><br />
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<span face="Helvetica, Verdana, Arial, sans-serif"><span style="line-height: 21px;">the 810nm wavelength can be used in existing fiber optic telecom technology, and the entangled nature of the photon opens up a range of ways to test QKD in existing equipment and infrastructure. </span></span></div><div style="list-style: none; margin-bottom: 15px; margin-left: 10px; padding: 0px;"><br /></div><div style="list-style: none; margin-bottom: 15px; margin-left: 10px; padding: 0px;"><br /></div><div style="list-style: none; margin-bottom: 15px; margin-left: 10px; padding: 0px;"><br /></div><div style="list-style: none; margin-bottom: 15px; margin-left: 10px; padding: 0px;"><br />
<span face="Helvetica, Verdana, Arial, sans-serif"><span style="line-height: 21px;">For example, using fibre optics it is possible to interface a telecoms 1550nm signal with a Lithium Niobate Electro-Optic Modulator (EOM). This can open up the possibility of having active system for performing measurements.</span></span><br />
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<span face="sans-serif" style="background-color: white; line-height: 19.2px;">If lithium niobate is exposed to an electric field, </span><span face="sans-serif" style="background-color: white; line-height: 19.2px;">created by placing a parallel plate capacitor</span><span face="sans-serif" style="background-color: white; line-height: 19.2px;"> across the crystal, </span><span face="sans-serif" style="background-color: white; line-height: 19.2px;">light will travel more slowly through it. The phase of the light leaving the crystal is directly proportional to the length of time it takes that light to pass through it. Therefore, the phase of the laser light exiting an EOM can be controlled by changing the electric field in the crystal.</span><br />
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<span face="sans-serif" style="background-color: white; line-height: 19.2px;">A phase modulating EOM can also be used as an amplitude modulator by using a </span>Mach-Zehnder interferometer<span face="sans-serif" style="background-color: white; line-height: 19.2px;">. A beam splitter divides the laser light into two paths, one of which has a phase modulator as described above. The beams are then recombined. Changing the electric field on the phase modulating path will then determine whether the two beams interfere constructively or destructively at the output, and thereby control the amplitude or intensity of the exiting light. This device is called a </span>Mach-Zehnder modulator<span face="sans-serif" style="background-color: white; line-height: 19.2px;">.</span><br />
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As discussed earlier, phase modulation using electro-optic modulators is more robust and does not disturb the polarization state, which can become randomized in an optic fiber over short distances. Hence, by having an active system of phase modulation we can have active QKD which is relatively stable.<br />
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Therefore Alice, the server, could in principle have several clients which share the common quantum key but which are free to modulate their individual signals via phase modulation, which does not disturb the polarization state of the entangled photon.<br />
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All of this is a way to test some basic principles of QKD with scalable equipment for applications in both fiber and free space communications<br />
<br /><br /><br /><br /></div><div style="list-style: none; margin-bottom: 15px; margin-left: 10px; padding: 0px;"><h2 style="text-align: left;"><u>Quantum Correlated Holography For Sub-Shot Noise Imaging</u></h2></div><div style="list-style: none; margin-bottom: 15px; margin-left: 10px; padding: 0px;"><br />
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The first photographs of downconversion light from a Beta Barium Borate (BBO) parametric downconversion crystal were take in the Innsbruck laboratory of the Institut für Experimantalphysik by Michael Reck and Paul Kwiat [<a href="http://www.tongue-twister.net/mr/physics/bbo_photo.htm#Reck" style="color: navy;">Reck 1996</a>]. They used high-speed infrared film and a 35mm single-lens reflex camera with the lens removed. The UV-light from the pump laser and fluorescence from the crystal were held back by stacks of UV-cutoff filters.</div>
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The camera was 11cm from the crystal with the UV-pump beam (wavelength 351nm) pointing to the center of the film. Interference filters with 5nm bandwidth selected a single color. A great number of photographs were taken on Kodak high-speed BW infrared film with different exposure times. The photographs were developed for 4min using Ilford Tech HC developer by Photo Grattl, Innsbruck. The optimal contrast on the film was achieved for exposure times of 1 hour at a UV-pump power of 165mW. The entangled photon pairs from this source were used to demonstrate a violation of Bell's inequalities by over 100 standard deviations in less than 5 min [<a href="http://www.tongue-twister.net/mr/physics/bbo_photo.htm#Kwiat" style="color: navy;">Kwiat 1995</a>].</div>
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<tr><td style="font-size: 10pt;" valign="TOP"></td><td style="font-size: 10pt;" valign="TOP"><b>Figure 1:</b> Schematic of the setup used to photograph type-II downconversion: The BBO crystal is pumped by an Argon ion laser with P=200mW at 351nm . An iris diaphragm helped to reduce background and reflected light. A tilted UV cutoff filter (UV-Sky F1) is used to reduce fluorescence from the exchangeable interference filter (IF2). We used 681nm, 702nm, 725nm interference filters with 5nm full-width-half-maximum (FWHM) bandwidth. A stack of cutoff filters (UVHaze F3, UVHaze F4, O2 F5) further reduce the background light. The camera is a Pentax K2 35mm single-lens reflex camera with the lens removed. The typical exposure time for the high speed infrared film was one hour.<br />
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<tr><td style="font-size: 10pt;" valign="TOP"><b>Figure 2:</b> Transmission curves of the cut-off filters used to photograph type-II downconversion from BBO.</td><td style="font-size: 10pt;" valign="TOP"><img align="BOTTOM" alt="Filters used in Setup" height="360" src="http://www.tongue-twister.net/mr/physics/bbo_filt.gif" width="500" /></td></tr>
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Original Infrared Film</h2>
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<b>Figure 3:</b> High speed infrared film exposed with light from type-II downconversion in BBO. A 681nm interference filter with 5nm bandwidth was used for this image.</blockquote>
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<b>Figure 4:</b> High speed infrared film exposed with light from type-II downconversion in BBO. A 725nm interference filter with 5nm bandwidth was used for this image.</blockquote>
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<b>Figure 5:</b> High speed infrared film exposed with light from type-II downconversion in BBO. A 702nm interference filter with 5nm bandwidth was used for this image. Polarization-entangled photons are observed at the intersection of the two circles.</blockquote>
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If we exploit the properties of quantum correlation of photons in entanglement, between two light beams, one sent to a physical object and the other kept as a reference, then it will also be possible to identify conditions that would be otherwise prohibitive due to classical constraints on sensitivity.<br />
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In this synthesis it is important to note that the number of photons of the single beam fluctuates randomly, but where the number of photons in the two beams fluctuates in unison.<br />
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The possible applications of the scheme are currently still hypotheticalexperimental, but range from from the identification of a poorly reflective object flying in the sky during the day, to the revelation of the presence of pollutants weakly reflective / absorbent in the atmosphere.<br />
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Experiments on "quantum illumination" has also demonstrated for the first time a protocol based on quantum correlations that can be efficient even in the presence of a noise (in the case even predominant): this represents a significant advance since all the protocols based on the entanglement achieved so far (from the transporter to ' imaging sub shot noise, from quantum computing to quantum cryptography) are extremely sensitive to the presence of noise, the presence of which quickly erases the benefits of quantum scheme. This result contradicts the opinion then, which was rooted, that quantum protocols are all extremely sensitive to noise and therefore difficult to apply in real-world conditions.<br />
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<i>Experimental scheme for '"imaging sub shot noise" using entangled photons from a parametric down-converter crystal. Such schemes could image objects impossible to image due to classical restraints on light reflection and transmission. Moreover, they could allow for quantum protocols to exist in the presence of environmental noise.</i><br />
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A proposal for testing this technology with a weather balloon and portable reflector telescope may also be a possibility with new versions of this equipment, which would be hindered if the detector or transmitter apparatus on board a weather balloon is too large. This opens up a possibility of having secure quantum communications and detection protocols in free space with fully portable infrastructure such as satellites, weather balloons or even manned and unmanned aerial vehicles.</div>
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<u><span style="font-size: large;">Hacking Quantum Signals - Stress test of QKD</span></u></h2>
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<span style="background-color: white;">However, it is important to bear in mind that quantum cryptography only protects transmissions. Alices and Bobs messages can still be tricked or coerced by inherent vulnerabilities in the detection system. Hence, conventional safeguards like passwords are still important. </span></div>
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<span style="background-color: white;">In practice, no quantum cryptographic system is perfect and errors will creep in owing to mundane environmental noise. Quantum physicists have calculated that as long as the mismatch between Alice's and Bob's keys is below low error threshold of ~8%, then security has not been breached.</span></div>
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<span style="background-color: white;"> <img alt="Falling numbers" src="http://www.nature.com/news/2010/100520/images/news.2010.quantum.cryptography.jpg" /></span></div>
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<span style="background-color: white;">Moreover, for certain QKD systems a hacker of quantum cryptography could disguise his interference as acceptable levels of noise, however as we have seen there are protocols being developed which can negate the effects of noise therfore the level of noise detectors will find acceptabe will always decrease. This is possible by having an arms race of sorts between the vendors of QKD equipment and people actively researching ways to make the most sensitive detectors possible which compensate noise, which is no easy task.</span></div>
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<span face="arial, helvetica, sans-serif" style="background-color: white; font-size: 13px; line-height: 18px;">A more prevalent threat is that Quantum key cryptography by polarization, has inherent vulnerabilities involving the key the sender "Alice" sends in the form of a series of polarized single photons to the receiver "Bob". </span><br />
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<span face="arial, helvetica, sans-serif" style="background-color: white; font-size: 13px; line-height: 18px;">Alice polarizes each photon at random using either a horizontal–vertical polarizer or a polarizer with two diagonal axes. Bob detects each photon by also randomly selecting one of the two different polarizers.</span><br />
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If Bob happens to pick the same polarizer as Alice, then he will definitely measure the correct polarization of a given photon. Otherwise, as the Heisenberg Uncertainty Principle dictates, there is a 50% chance he will get it wrong. Once he has made all the measurements, Bob asks Alice over an open channel which polarizers she used for each photon and he only keeps the results for those measurements where he happened to pick the correct polarizer, and this series of results becomes the secret key.</div>
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Meanwhile, an eavesdropper, "Eve", who seeks to measure the polarization of the photons sent by Alice would reveal her presence because, given a long enough string of photons, the probability of her correctly guessing Alice's sequence of polarizers becomes practically zero. </div>
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When she makes incorrect measurements, she randomizes the polarization. This means that in some of the cases where Bob should make a correct measurement, he makes a wrong one. So, again speaking openly with Alice and comparing a small subset of the key, Bob realizes there is an intruder if that subset contains errors.</div>
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<span face="arial, helvetica, sans-serif" style="font-size: 13px; line-height: 18px;">There is however a way to hide Eve's eavesdropping by exploiting a weakness in the single-photon detectors used in many commercially available quantum-cryptographic receivers. This involves Eve using a bright laser light to "blind" the four avalanche photodiodes that Bob uses to detect photons in each of the four different polarization states. </span><span face="arial, helvetica, sans-serif" style="font-size: x-small;"><span style="line-height: 18px;">Eve, therefore must contain a Bob', to detect the photons, and Alice', to generate a photon with the same polarization as the detected photon, where the output of Alice' must be coupled, via and optical amplifier, to the blinding laser</span></span></div>
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The laser-blinded photodiodes are no longer sensitive to single photons, but instead behave like classical detectors that generate a current proportional to the intensity of the incoming light and respond to pulses of light above a certain intensity threshold.</div>
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<a href="https://encrypted-tbn1.gstatic.com/images?q=tbn:ANd9GcRsH1sk8C_8RYNOeXhdpvqwtsSRchZw7Krope8BOCY2MNDuBeinyw" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="190" src="https://encrypted-tbn1.gstatic.com/images?q=tbn:ANd9GcRsH1sk8C_8RYNOeXhdpvqwtsSRchZw7Krope8BOCY2MNDuBeinyw" width="320" /></a>Therefore, Eve intercepts each of the photons sent by Alice and measures them using randomly chosen polarizers. With each measurement Eve sends a bright pulse of light, above the intensity threshold and with the same polarization as the photon measured, to Bob's detectors. This removes Bob's ability to randomly assign polarizers for each measurement. Instead he is constrained to the same sequence of polarizations as obtained by Eve. This means that when Bob and Alice publicly compare the subset of the key, they find no errors. In other words, Eve has found out the key and has remained hidden while doing so.</div>
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<span face="arial, helvetica, sans-serif" style="font-size: 13px; line-height: 18px;">In the case of phase controlled QKD the spatio-temporal mismatch created by the optical amplifier could be detected by Bob by means of an interferometer, as the </span><span face="arial, helvetica, sans-serif" style="font-size: x-small;"><span style="line-height: 18px;">bit values are encoded in the phases of individual photons.</span></span><br />
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<span face="arial, helvetica, sans-serif" style="font-size: x-small;"><span style="line-height: 18px;">The interaction of intense laser light can lead to the re-absorption of previously generated photons, depending on the relative phase between the two. Different phase velocities lead to destructive interference due to the lack of optical momentum conservation between the photons, known as “phase mismatch”.</span></span><br />
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<span face="arial, helvetica, sans-serif" style="font-size: x-small;"><span style="line-height: 18px;"><i>Left: Traditional concept of spatial phase mismatch. </i></span></span><br />
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<a href="http://www.mics.caltech.edu/graphics/pingPongCDR.png" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="http://www.mics.caltech.edu/graphics/pingPongCDR.png" style="cursor: move;" width="340" /></a>For sufficient phase mismatch between signal paths 2 independently adjustable clock phases for Alice and Bob are generated from a calibrated delay line. </div>
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Bob's clock phase is placed in the middle of the eye to recover the data, while Alice's is swept across the delay line. The samples produced by the two clocks are then compared, by classical means, to generate eye information, which is then used to determine the best phase for data recovery. The functions of the two clocks are swapped after the data phase is updated; this ping-pong action allows an infinite delay range and thus would make an infinitely large number of ranges for a hacker to chose a phase for silent data recovery, hence negating any (known) means for attack.</div>
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<div style="font-family: Arial, Verdana, Helvetica, sans-serif; line-height: 17.92px;"><span style="background-color: white;"><br /></span></div><div style="font-family: Arial, Verdana, Helvetica, sans-serif; line-height: 17.92px;"><br /></div><div style="font-family: Arial, Verdana, Helvetica, sans-serif; line-height: 17.92px;"><br /></div><div style="font-family: Arial, Verdana, Helvetica, sans-serif; line-height: 17.92px;"><span style="background-color: white;">Using the Quantum Random Number Generator however as an entropy source for a backup key is useful as it depends on the anti-correlations.</span></div><div style="font-family: Arial, Verdana, Helvetica, sans-serif; line-height: 17.92px;"><span style="background-color: white;"><br /></span></div><div style="font-family: Arial, Verdana, Helvetica, sans-serif; line-height: 17.92px;"><span style="background-color: white;"><br /></span></div><div style="font-family: Arial, Verdana, Helvetica, sans-serif; line-height: 17.92px;"><span style="background-color: white;"><br /></span></div><div style="font-family: Arial, Verdana, Helvetica, sans-serif; line-height: 17.92px;"><span style="background-color: white;"><br /></span></div><div style="font-family: Arial, Verdana, Helvetica, sans-serif; line-height: 17.92px;"><span style="background-color: white;"><br /></span></div><div style="font-family: Arial, Verdana, Helvetica, sans-serif; line-height: 17.92px;">
<span style="background-color: white;">Any investments made by governments and corporations into this new technology will be guaranteed by the laws of physics, but only as much as we understand them. </span><br />
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<span style="background-color: white;">Hopefully with more stress tests we will continue onward, improving on design and discovery and implementing this technology with the best intentions, trying to bring about a more connected, but more secure, global civilization.</span><br />
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Long exposure photography reveals some great detail of the galactic dust lanes containing the opaque cosmic dust that forms filaments in the plane of the galactic disks. </span></div><div><span style="background-color: white; color: #0d0d0d; font-family: Roboto, Noto, sans-serif; font-size: 15px; white-space: pre-wrap;"><br /></span></div><div><span style="background-color: white; color: #0d0d0d; font-family: Roboto, Noto, sans-serif; font-size: 15px; white-space: pre-wrap;">T</span><span style="background-color: white; color: #0d0d0d; font-family: Roboto, Noto, sans-serif; font-size: 15px; white-space: pre-wrap;">hese dark regions of the sky, in contrast with the bright star fields concentrated in the spiral arms, create distinct shadowy shapes that were interpreted as constellations in their own right along with the connect-the-dots style constellations of the ancient Inca Civilization in particular. </span></div><div><span style="background-color: white; color: #0d0d0d; font-family: Roboto, Noto, sans-serif; font-size: 15px; white-space: pre-wrap;"><br /></span></div><div><span style="background-color: white; color: #0d0d0d; font-family: Roboto, Noto, sans-serif; font-size: 15px; white-space: pre-wrap;">Software used is Sequator, a great piece of open-source, regularly updated software that deserves support! check it out: </span><span style="background-color: white; color: #0d0d0d; font-family: Roboto, Noto, sans-serif; font-size: 15px; white-space: pre-wrap;">https://sites.google.com/site/sequatorglobal/download</span></div>MuonRayhttp://www.blogger.com/profile/03712859045968965104noreply@blogger.com0tag:blogger.com,1999:blog-5996268492156364201.post-60752264557979283052021-10-29T14:04:00.001-07:002021-10-29T14:04:24.421-07:00Drone Astrophotography at Astronomy Ireland Star-B-Q 2021 | Orion and Or...<iframe frameborder="0" height="270" src="https://youtube.com/embed/1Wiosz3K_vQ" width="480"></iframe><div><br /></div><div>Livestream Starting at 9:45pm GMT. </div>MuonRayhttp://www.blogger.com/profile/03712859045968965104noreply@blogger.com0tag:blogger.com,1999:blog-5996268492156364201.post-31396893930134095752021-10-03T10:34:00.006-07:002021-10-25T07:12:03.047-07:00Denoising Drone Images with the ROF Algorithm in Python | For NDVI, UV a...<iframe frameborder="0" height="270" src="https://youtube.com/embed/lDAoHecn-2Q" width="480"></iframe><div><br /></div><div><br /></div><div><div>Consider Donating to Help the development of this project further:https://www.paypal.com/biz/fund?id=BLJ283JMTMT7S</div><div><br /></div><div>Coding Used in This Video Here:https://github.com/MuonRay/Image_Denoising_with_ROF_algorithm</div><div><br /></div><div>Removing noise from images is important for many applications, from making personal photos look better to improving the quality of satellite and increasingly drone images.</div><div><br /></div><div>In image processing denoising functionally looks like we are smoothing out the image. But just what is it we are smoothing out to remove the noise? </div><div><br /></div><div>The ROF model has the interesting property that it finds a smoother version of the image while preserving edges and structures.</div><div><br /></div><div>The underlying mathematics of the ROF model and the solution techniques are quite advanced and are showcased fully in the paper:</div><div><br /></div><div>Nonlinear total variation based noise removal algorithms*</div><div>Leonid I. Rudin 1, Stanley Osher and Emad Fatemi </div><div>Cognitech Inc., 2800, 28th Street, Suite 101, Santa Monica, CA 90405, USA. circa 1992. </div><div><br /></div><div>Its interesting to note that the authors base their work on work done previously by Geman and Reynolds in which they propse to minimise the non-linear functionals associated with noise in the total variance by use of simulated annealing, a metaheursitics technique, which would have been very slow to do using the computers available in the late 1980s something</div><div>which drove the development of the ROF denoising model in the first place.</div><div><br /></div><div>The authors wanted a fast solver that could find a reasonably good local minima rather than the ideal global minima. </div><div><br /></div><div>Here I'll give a brief, simplified introduction before showing how to implement a ROF solver based on an algorithm by Chambolle.</div><div><br /></div><div>The solver used in this code is a modified version that uses gradient descent/reprojection method to achieve total variance (TV) minimization/regularization</div><div><br /></div><div>The integral of the gradient across an image, any image, will produce the total varience in the image. now, for noisy images the total varience will be higher. knowing this, denoising techniques have been developed that essentially minimise the total varience in the matrix element of an image and then reproject that image onto the original by subtracting</div><div>the imaginary form of the original image matrix element which contains the residual texture of the image. so textures are effectively removed when we want to do TV minimization/regularization.</div><div><br /></div><div><br /></div><div><br /></div><div>To minimise the total varience in the matrix element different algorithms can be used but one of the most popular is gradient descent, which is similar to the simulated annealing technique originally proposed but more computationally tractable.</div><div><br /></div><div><br /></div><div>in gradient descent the image matrix containing the greyscale pixel values are essentially represented as an energy surface, whereby </div><div>we want to descend into the global minima. the different values in the matrix represent the interaction energies of the nearest neighbor in this 2D energy surface. </div><div>different algorithms can use</div><div>different equations to represent the interaction energies, depending on the rate at which</div><div>the interaction energies converge to a global minima in a certain steplength of </div><div>iteration of the algorithm. </div><div><br /></div><div><br /></div><div><br /></div><div>Interesting Note: When implented using periodic boundary conditions the TV minimization using </div><div>an iteractive gradient descent on an image performs 2 transformations on that</div><div>images rectangular image domain where the greyscale pixel values exist. 2 transformations </div><div>on a rectangle result in the formation of a torus in a transformation which </div><div>preserves the images pixel data but changes the domain topology. </div><div> </div><div> </div><div> </div><div> </div><div> An implementation of the Rudin-Osher-Fatemi (ROF) denoising model</div><div> using the numerical procedure presented in equation 11 on pg 15 of</div><div> A. Chambolle (2005)</div><div> http://www.cmap.polytechnique.fr/preprint/repository/578.pdf</div></div>MuonRayhttp://www.blogger.com/profile/03712859045968965104noreply@blogger.com0tag:blogger.com,1999:blog-5996268492156364201.post-7254087938971353492021-09-06T16:59:00.000-07:002021-09-06T16:59:41.798-07:00An Aerial Tour of Gran Canaria - The "Mini Continent" Island of Contrasts at Every Visual Angle<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhwB8cD8k7vt-BW0VOO-rzRor_rz4PBDWf74bxrgdAm3M1dymvVqUJZKOmjoKG3iyj7UdRdxJwczFfw92Ic4nZ78pLuCu84JOtZBrEQ966gYxqbs83mDcUq5jEisGx8FKP-2kBaNmYLCrBW/s648/gran+canaria+satellite+map+gif+animation.gif" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="648" data-original-width="648" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhwB8cD8k7vt-BW0VOO-rzRor_rz4PBDWf74bxrgdAm3M1dymvVqUJZKOmjoKG3iyj7UdRdxJwczFfw92Ic4nZ78pLuCu84JOtZBrEQ966gYxqbs83mDcUq5jEisGx8FKP-2kBaNmYLCrBW/s320/gran+canaria+satellite+map+gif+animation.gif" width="320" /></a></div><br /><div>Gran Canaria is the capital island of the archipelago of islands known as The Canaries. While not the largest island, nor the most visited by tourists, in many respects it hosts the most in terms of diversity and this is perhaps the result of, or the reason for, the relatively large influence this one island has among its neighbors. </div><div><br /></div><div>Having spent almost a year working there and exploring the island extensively on hiking trails and climbing sites I was given ample opportunity to deploy my drone to do environmental and geological investigation from the air, using true colour and near-infrared imaging to gain an extra appreciation for the diverse landscapes of the island. </div><div><br /></div><div>As I found out, and you may see, it is an island of many different parts; containing jungles, forests, canyons, volcanic craters, deserts along with beaches, rocky and sandy, and urban environments not to mention marine and freshwater habitats. All operating in tandem to support multiple distinct microbiomes. </div><div><br /></div><div>I have compiled the highlights of these trips into a single video and have also created a series of individual videos, in many cases comparing how the natural features look in the visible, near-infrared and in some cases near-ultraviolet spectrum.</div><div><br /></div><div>This collection of footage helped me to create several short films showcasing topics in science and technology, the 4K and RAW format images taken during these trips into the wild areas of Gran Canaria are also valuable in the development of new remote sensing techniques using drone imaging. </div><div><br /></div><h3 style="text-align: left;"><u>Drone Footage Compiled into a Single Video:</u></h3><div><br /></div><div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen="" class="BLOG_video_class" height="266" src="https://www.youtube.com/embed/GUP7g8Tlx98" width="320" youtube-src-id="GUP7g8Tlx98"></iframe></div><br /><div><br /></div><div><br /></div><div><br /></div><div>Region-specific drone flight footage (for more detail):</div><div><br /></div><h3 style="text-align: left;"><u>Las Palmas during the Day:</u></h3><div><br /></div><div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen="" class="BLOG_video_class" height="266" src="https://www.youtube.com/embed/VgNXx0W9z1U" width="320" youtube-src-id="VgNXx0W9z1U"></iframe></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><br /></div><h3 style="text-align: left;"><u>Las Palmas at Night:</u></h3><div><br /></div><div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen="" class="BLOG_video_class" height="266" src="https://www.youtube.com/embed/C7zBoCACWbo" width="320" youtube-src-id="C7zBoCACWbo"></iframe></div><br /><div><br /></div><h3 style="text-align: left;"><u>El Confittal:</u></h3><div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen="" class="BLOG_video_class" height="266" src="https://www.youtube.com/embed/kj8-ZxKcp5A" width="320" youtube-src-id="kj8-ZxKcp5A"></iframe></div><div><br /></div><div><br /></div><h3 style="text-align: left;"><u>Barranco Los Tilos de Moya:</u></h3><div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen="" class="BLOG_video_class" height="266" src="https://www.youtube.com/embed/z6JO_FORfHg" width="320" youtube-src-id="z6JO_FORfHg"></iframe></div><br /><div><br /></div><h3 style="text-align: left;"><u>Roque Nublo:</u></h3><div><div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen="" class="BLOG_video_class" height="266" src="https://www.youtube.com/embed/AVNLCJ7NfjE" width="320" youtube-src-id="AVNLCJ7NfjE"></iframe></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><br /></div><h3 style="text-align: left;"><u>Puerto Rico - Barranco de Lechugal:</u></h3><div><div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen="" class="BLOG_video_class" height="266" src="https://www.youtube.com/embed/hl1iD3Bm5m0" width="320" youtube-src-id="hl1iD3Bm5m0"></iframe></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><br /></div><br /><h3 style="text-align: left;"><u>Bandama Caldera:</u></h3></div><div><div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen="" class="BLOG_video_class" height="266" src="https://www.youtube.com/embed/p68s5qR6nCA" width="320" youtube-src-id="p68s5qR6nCA"></iframe></div><div class="separator" style="clear: both; text-align: center;"><br /></div><br /><h3 style="text-align: left;"><u>Dunas Maspalomas: </u></h3></div><div><div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen="" class="BLOG_video_class" height="266" src="https://www.youtube.com/embed/1sHexwD7AKw" width="320" youtube-src-id="1sHexwD7AKw"></iframe></div><br /><u><br /></u></div><h3 style="text-align: left;">Barranco Del Toro - Featured In a Short Nature Film I made on the Convergent Evolution of the Native Euphorbia Canariensis in comparison to the Artificially Introduced "True Cacti" (<span style="background-color: #202124; font-family: arial, sans-serif; font-size: 14px;"><span style="color: #04ff00;">Cactaceae</span></span>) and how they endure in the desert habitat of the Canary Islands.</h3><div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen="" class="BLOG_video_class" height="266" src="https://www.youtube.com/embed/RqsuK5R9quY" width="320" youtube-src-id="RqsuK5R9quY"></iframe></div><br /><div><br /></div><u><br /></u></div><div><br /></div>MuonRayhttp://www.blogger.com/profile/03712859045968965104noreply@blogger.com0tag:blogger.com,1999:blog-5996268492156364201.post-37419510997041820182021-08-07T15:15:00.002-07:002021-08-16T08:38:08.385-07:00Drone Astrophography Imaging using Python and Sequator - Viewing The Andromeda Galaxy!🌌<p>In this video I show how I was able to take some interesting deep sky astrophotography images using my Drone Camera (a modified Hasselblad 20MP Camera onboard a Mavic 2 Pro) in which one can see, among other things, the Andromeda Galaxy. </p><div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen="" class="BLOG_video_class" height="299" src="https://www.youtube.com/embed/lPAoo1uxLLE" width="560" youtube-src-id="lPAoo1uxLLE"></iframe></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div>This is made possible using coding I had been experimenting with in Python (available here:https://github.com/MuonRay/Python_DNG_Drone_Astrophotography) and the exposure stacking software Sequator (available here:https://sites.google.com/view/sequator/download)</div><div><br /></div><div>This video is a showcase of ongoing experiments I am carrying out to test the limits of the "professional/consumer" grade of drones currently available and find out what exactly they are fully capable of doing in the context of using them for Earth Observation and Night Sky observation. So far I have been very impressed with the results so far and feel I've only scratched the surface. </div><div><br /></div><div>The fact that the drone can take better astrophotography images while flying than stationary on the ground was a pleasant surprise to me and opens up the possibility of detailed night time stargazing using drones perhaps to accompany ground based observations of the sky.</div><div><br /></div><p>August 13th update:</p><p>I tried some Meteor Tracking using my Drone Near-Infrared Moodified Camera and by taking a few hundred images and combining them into a video I managed to get some interesting results from the Perseid Meteor Shower:</p><div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen='allowfullscreen' webkitallowfullscreen='webkitallowfullscreen' mozallowfullscreen='mozallowfullscreen' width='320' height='266' src='https://www.blogger.com/video.g?token=AD6v5dzgE1U7CQ69j349zvGL8sbUiJtjho2A9OS_N5Km4yGXPds0Qk_Duyd1_g9OIOGSdG06WnDaNUs-QgbbewuWlQ' class='b-hbp-video b-uploaded' frameborder='0'></iframe></div><br /><p><br /></p>MuonRayhttp://www.blogger.com/profile/03712859045968965104noreply@blogger.com0tag:blogger.com,1999:blog-5996268492156364201.post-44969069245707363752021-07-23T16:12:00.002-07:002021-07-27T09:42:55.012-07:00Ultraviolet Reflectance Photography with Drone Imaging Update - A Near-UV Reflectance Index (UVRI) for Flowering Plant Imaging<p>Using my Multi-spectral converted Hasselblad Camera onboard the DJI Mavic 2 Pro I have been developing an experimental UV Reflectance Index program using UV-Pass Filter </p><p>I have developed image processing codes that work with 16-bit DNG and JPG image files which creates the reflectance index that highlights the flowers, in particular those which are strongly fluorescent.</p><p><a href="https://github.com/MuonRay/Ultraviolet_Image_Python_Processing_Codes">https://github.com/MuonRay/Ultraviolet_Image_Python_Processing_Codes</a></p><p><br /></p><p>Using these test images:</p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjdcCy_anSGPi-8IgnbTnJGKDTqmDXhff8i4cWna2QQbaoESc0bbnijjbF4ngF2s40QQGpbtQG3LUYzUT8CliNaxx2DPh3nTKeGU0tu-M1UkEErULrBkiMd4aDVOtdqsaiRO4UL35Ampxir/s5472/DJI_0210.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="3078" data-original-width="5472" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjdcCy_anSGPi-8IgnbTnJGKDTqmDXhff8i4cWna2QQbaoESc0bbnijjbF4ngF2s40QQGpbtQG3LUYzUT8CliNaxx2DPh3nTKeGU0tu-M1UkEErULrBkiMd4aDVOtdqsaiRO4UL35Ampxir/s320/DJI_0210.JPG" width="320" /></a></div><br /><p></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgMoR3rqkFfim4NLTj44AJPA2cj1fovhxl2PXnM3P3DdFxQdsYK2Jg80y84AEPFYoLxo8nxpiO2gu-CM7ntt60OjKmmuyOoMwjKT1GQhg51Nf_PDkVMG7uV2i-arrkD7DKmuqbLRDH9nIba/s5472/DJI_0213.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="3078" data-original-width="5472" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgMoR3rqkFfim4NLTj44AJPA2cj1fovhxl2PXnM3P3DdFxQdsYK2Jg80y84AEPFYoLxo8nxpiO2gu-CM7ntt60OjKmmuyOoMwjKT1GQhg51Nf_PDkVMG7uV2i-arrkD7DKmuqbLRDH9nIba/s320/DJI_0213.JPG" width="320" /></a></div><p><br /></p><p>The following UVRI are created:</p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgObhYEAuQ1G9_7ZOndAAHr3c_JN9TTm56OrYxx5iAKwUSnYpQm_Ws29-OSpSpm9i0TI1fQajjIaGqUMoYzAng9PYZ7Qjr34xfnGdEGeD7bfppw-A7AzuraSHeqLIatbch5ws-Uq3o8tYn4/s2048/UVReflectanceIndex2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1330" data-original-width="2048" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgObhYEAuQ1G9_7ZOndAAHr3c_JN9TTm56OrYxx5iAKwUSnYpQm_Ws29-OSpSpm9i0TI1fQajjIaGqUMoYzAng9PYZ7Qjr34xfnGdEGeD7bfppw-A7AzuraSHeqLIatbch5ws-Uq3o8tYn4/s320/UVReflectanceIndex2.jpg" width="320" /></a></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgPGY8vlj4gaHfW0dGmO5_JDt_e8zi1Iv0M5Ed0Xbo-TRIGyUzsTHm57QlVAH6JPGZjh2iaSkzViqcEhsykYCU9g6ZiEDrb-FKx0W-t3qF3noL24jArYex9_yq1yV3_XcS6y6agrpNptArg/s2048/UVReflectanceIndex.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1330" data-original-width="2048" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgPGY8vlj4gaHfW0dGmO5_JDt_e8zi1Iv0M5Ed0Xbo-TRIGyUzsTHm57QlVAH6JPGZjh2iaSkzViqcEhsykYCU9g6ZiEDrb-FKx0W-t3qF3noL24jArYex9_yq1yV3_XcS6y6agrpNptArg/s320/UVReflectanceIndex.jpg" width="320" /></a></div><br /><p><br /></p><p>This work is the next stage in my own research in the use of drones for remote sensing for environmental monitoring and plant health sensing, following on from developments made in the Near-Infrared which allow for NDVI map making using drone photogrammetry. </p><p>Next steps in this work will be to expand the image processing for novel applications of the UVRI images for potential flower classification.</p><p><br /></p><p><br /></p><br />MuonRayhttp://www.blogger.com/profile/03712859045968965104noreply@blogger.com0tag:blogger.com,1999:blog-5996268492156364201.post-6976304942118967962021-05-22T12:50:00.003-07:002021-06-14T17:43:06.775-07:00Quantum Foundations of Reality: A Physics Talk on Condensates, Chaos, Chimera States and Computing.<iframe frameborder="0" height="270" src="https://youtube.com/embed/WX3Ds_xCOaE" width="480"></iframe><div><br /></div><div><span style="background-color: white; color: rgba(0, 0, 0, 0.87); font-family: Roboto, Noto, sans-serif; font-size: 15px; white-space: pre-wrap;">A documentary/lecture I have compiled using previous videos in which I discuss the nature of quantum phenomena, the principles which govern their behavior and how we can use mathematical physics models to understand how these effects can work across the scales of reality, creating the emergent nature of chaos and chimera states in both quantum and classical systems, in a sense unifying them. </span><span style="background-color: white; color: rgba(0, 0, 0, 0.87); font-family: Roboto, Noto, sans-serif; font-size: 15px; white-space: pre-wrap;">
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</span><span style="background-color: white; color: rgba(0, 0, 0, 0.87); font-family: Roboto, Noto, sans-serif; font-size: 15px; white-space: pre-wrap;">The introduction is taken from an old interview with American Physicist John Archibald Wheeler in which he discusses the mysterianism inherent in the quantum mechanical nature of our reality in particular with regard to the observer effect and how the quantum phenomena are only made manifest in our reality by the act of observation which transform the infinite probabilities, a "smoky dragon" as it were, into a real and definite manifestation. </span><span style="background-color: white; color: rgba(0, 0, 0, 0.87); font-family: Roboto, Noto, sans-serif; font-size: 15px; white-space: pre-wrap;">
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</span><span style="background-color: white;"><span style="color: rgba(0, 0, 0, 0.870588235294118); font-family: Roboto, Noto, sans-serif;"><span style="font-size: 15px; white-space: pre-wrap;">The very fact that systems of quantum interactions can be modeled using classical analogies and these models themselves give rise to descriptions of collective quantum behavior also creates an added layer of mysterianism to the quantum nature of the universe. This is the emergent nature of the quantum mechanical equivalent of chaos and how ensembles of quantum particles modeled as oscillators will create unique behavior in the meta-stable medium between ordered interactions and randomization. </span></span></span><span style="background-color: white; color: rgba(0, 0, 0, 0.87); font-family: Roboto, Noto, sans-serif; font-size: 15px; white-space: pre-wrap;">
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</span><span style="background-color: white; color: rgba(0, 0, 0, 0.87); font-family: Roboto, Noto, sans-serif; font-size: 15px; white-space: pre-wrap;">These descriptions model not only how quantum systems behave in isolation but how they interact with the external forces that transform them from "smokey dragons" into definite phenomena. </span><span style="background-color: white; color: rgba(0, 0, 0, 0.87); font-family: Roboto, Noto, sans-serif; font-size: 15px; white-space: pre-wrap;">
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</span><span style="background-color: white; color: rgba(0, 0, 0, 0.87); font-family: Roboto, Noto, sans-serif; font-size: 15px; white-space: pre-wrap;">In effect we can view quantum mechanics itself as an inherent, scale dependent factor of the entire universe that operates in a balance between utilization of certain transition rules with the intrinsic flow of randomization that permeates the entire universe. </span><span style="background-color: white; color: rgba(0, 0, 0, 0.87); font-family: Roboto, Noto, sans-serif; font-size: 15px; white-space: pre-wrap;">
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</span><span style="background-color: white; color: rgba(0, 0, 0, 0.87); font-family: Roboto, Noto, sans-serif; font-size: 15px; white-space: pre-wrap;">Even more astonishing is just how many so-called "classical" phenomena also behave this way, especially when considering collective and complex adaptive systems. This more than any other reason is why it makes sense to begin to solve many of these multifaceted problems through the lens of quantum mechanics, a lens in which the emergent field of quantum computation will be the key to the entire enterprise of simulation and optimization of scenarios throughout the scale of reality.</span></div>MuonRayhttp://www.blogger.com/profile/03712859045968965104noreply@blogger.com0tag:blogger.com,1999:blog-5996268492156364201.post-47069063973533393212021-04-29T11:24:00.005-07:002021-06-14T17:54:02.868-07:00Quantum Chimera States - Unifying Chaos, Synchronization and Computation in Complex Systems Physics<iframe frameborder="0" height="270" src="https://youtube.com/embed/9K6KqybDJds" width="480"></iframe><div><br /></div><h2 style="text-align: left;">Introduction</h2><div><span face="Roboto, Arial, sans-serif" style="background-color: white; color: #030303; font-size: 14px; white-space: pre-wrap;">Complex systems have many behaviors that operate independently across scales. One of the universal phenomena of complex systems are the so-called Chimera States, the metastable states that exist in between the states of chaos and order, in systems of classical and quantum oscillators.
Chimera states emerge in many different domains of complex systems science, from biological synchronization, the physics of quantum phase transitions, the factors that influence social dynamics and the patterns that exist in so-called emergent phenomena in many different domains of reality.
Taking its name from the mythological beast from Greek Mythology, the Chimera is a fusion of beings that manifests from chaos, is difficult to track and disappears when being examined or pursued by a hunter. This legend has an ironic resonance when we use this creature as an analogy of the fusion of order and disorder in systems which can be treated as systems of oscillators.
The main system of oscillators we are concerned with are the so-called pure quantum systems that we have discussed in previous videos, which obey precise and ordered quantum transition rules but nevertheless have chaotic behavior.
Due to the universality of the chimera states across many different fields of study, from quantum to classical systems, the pure quantum systems being used in networks for computation and optimization problems offer a way to simulate and solve many equivalent behaviors in the real world. Its an astonishing but true fact that many of the behaviors of chimera states are scale independent and operate in complex classical, quantum and even relativistic systems.
Not taking this amazement for granted we may have to think about many of the seemingly separate fields of science as being under the shadow of the underlying quantum chimera. This may be the key to solving many of the recalcitrant problems in the world today and allow us to reach new levels of understanding and emulation with computer technology built around the operation and use of chimera states.</span><span face="Roboto, Arial, sans-serif" style="background-color: #f9f9f9; color: #030303; font-size: 14px; white-space: pre-wrap;">
</span></div><div><br /></div><h1 style="text-align: left;">The Quantum Chimera:</h1><div><br /></div><div><div>The observable universe hosts a vast array of complex systems across many different fields of interest.</div><div>These can be modelled using theories that create a layer of abstraction to separate the complexity from the </div><div>fundamental nature of the individual interactions. </div><div><br /></div><div>Describing complex systems as "systems of oscillators" may sound obscure, but in a strange convenience it describes, in a very general way, </div><div>all sorts of physical systems not just in the context of standard classical and quantum harmonics.</div><div><br /></div><div>As examples, there are lots of biological systems can be reduced to populations of harmonic and anharmonic oscillators. </div><div>The heartbeat is just oscillating heart cells that a wave propagates on. </div><div>And synchronizing neurons in the brain are oscillators as well, and have been treated with these methods to give rich understanding </div><div>of the kinds of patterns we see experimentally.</div><div><br /></div><div>Rather than just simply observing and recording the patterns from such reductionist systems, abstract generative models must be constructed if we are to enrich our </div><div>understanding beyond simply measuring outputs from abstract models. </div><div><br /></div><div>In higher level modelling we can say that Ensembles of globally coupled systems of oscillators synchronization can manifest</div><div>themselves into the appearance of a macroscopic mean field. In quantum field theory therefore an ensemble of particles, highly coupled through entanglement say</div><div>, can give rise to a scalar field or a scalar field can itself can mediate a strong coupling between ensembles of particles. Both Views happen to be equivalent</div><div>or have what we call a duality of description.</div><div><br /></div><div>Natural systems of synchronized particles can form macroscopic mean fields, such as atoms in Bose-Einstein Condensates or electrons in Superconductors.</div><div><br /></div><div><br /></div><div>In a condensate we see spontaneous breaking of symmetry of the particle where it does not matter if one adds a particle or subtracts it from the condensate, </div><div>a mechanism which may appear odd in terms of charge transfer but in terms of representing this in terms of momentum it appears obvious as in the newtons cradle </div><div>model.</div><div><br /></div><div>The core reason why we have the separation of the different nodes in an ensemble of particles is due to the uncertainty principle </div><div>and in condensates of bosons this is the cause of the separation that we see.</div><div><br /></div><div>Quite often clusters of synchronized elements are observed in between regions of unsynchronized elements.</div><div>In a type-II superconductor we see an intermediate phase of ordinary conductivity by unsynchronized electrons </div><div>mixed with superconductivity mediated by the synchronization of electrons in cooper pairs. </div><div><br /></div><div>this effectively means we have a mixture of a synchronous condensate with asynchronous individual charge carriers</div><div>at intermediate temperature and fields above the superconducting phases. </div><div><br /></div><div>Synchronous and asynchronous might seem dichotomous conditions of a functioning system, yet both states can, in fact, </div><div>exist simultaneously and durably within a system of oscillators, in what's called a chimera state. </div><div><br /></div><div>Chimera states are patterns where synchronous and asynchronous domains coexist, taking its name from a composite creature in Greek mythology. </div><div>According to the myth, the main power of the chimera is that the closer someone gets to it, as they are pursuing it, The further away it is from you in actuality.</div><div><br /></div><div>This has an ironic resonance with the chimeras we are talking about in the context of systems of quantum oscillators and chaos theory.</div><div>This exotic state still holds a lot of mystery, but its fundamental nature offers potential in understanding governing dynamics across many scientific fields.</div><div><br /></div><div>Of particular interest in this work is the effect symmetries of a complex system can have on the emergence of chimera states. </div><div><br /></div><div>for example, the effect of having the same versus different coupling strengths of the </div><div>outer regions of the field to the center regions. this causes breaking in the symmetry of synchronization and therefore drives the </div><div>system to adapt and perform a kind of emergent error correction or annealing. Such annealing can occur in systems of harmonic oscillators, </div><div>classical and quantum. Chimera states are therefore an integral part of any emergent complex adaptive system. Such systems can</div><div>be used to simulate other complex systems with equivalent behaviour and can be used to solve optimization problems.</div><div><br /></div><div>Several optimization problems can be represented as paths or logical decision trees, which themselves </div><div>can be reduced to the Boolean satisfiability problem, or SAT problem. Basically, it's an algebraic or Boolean logic expression </div><div>(that looks like ^ means AND, v means OR,means NOT)</div><div><br /></div><div>Think for example of trying to find the shortest path on a map from one city to another. </div><div>If you wanted to do a brute-force search, you'd be trying every single possibility that existed. </div><div>Another example is the knapsack problem, where you have a bunch of items but you can only carry a certain amount. </div><div><br /></div><div>Suppose you're working for NASA and you're building a rocket, and you're trying to figure out how much fuel to put on the rocket. </div><div>But remember fuel adds weight, so what are the optimal fuel to payload ratios? </div><div>What's the value of each item versus its weight where there are n objects with weights w_i and values v_i? </div><div><br /></div><div>Or Perhaps you are studying protein folding, and you need to fold in the minimum energy conformation </div><div>so that you can understand how diseases work. Now in these scenarios you might also realize nature doesn't always trend towards the global minimum energy conformation, </div><div>and asynchronous affects have to be considered existing with synchronous effect so we are already entering the territory so to speak of the chimera.</div><div><br /></div><div>Because this is an optimization problem, the notion of solution is not entirely adequate, backtracking is rather designed for decision problems, </div><div>in which one should answer questions of the type "is there a feasible solution?" or "is there a feasible solution achieving a value of at least V?". </div><div>In this case the extension of a solution might again be a solution.</div><div><br /></div><div>These are all massive problems with tons of simulated parameters but the common The goal is to maximize the value of the objects selected, </div><div>respecting a limit W on the sum of the weights of the selected objects. </div><div><br /></div><div>Using the Adiabatic Theorem in the design of a Quantum Network can be primarily used to solve the Boolean satisfiability problem, </div><div>like the clique network problem, a clique being a complete subgraph of a total graph. The size of a clique is the number of vertices it contains. </div><div>The clique problem is the optimization problem of finding a clique of maximum size in a graph. </div><div>Using the adiabatic Hamiltonian, where the solutions are the maxima and minima in </div><div>terms of a graph where there is a collection of nodes in a grid. This graph is analogous to a "program" in an AQC, </div><div>where the initial state of qubits are connected in a certain way.</div><div><br /></div><div>As a side note it so happens that even though adiabatic annealers and gate-based quantum computers are vastly different paradigms, </div><div>the Wigner Jordan transformation allows one to map fermionic problems onto an Ising spin model. </div><div>This spin model can of course be implemented on an annealer, or through the use of a Variational Quantum Algorithm, be implemented on a gate based computer. </div><div>The "annealing" part of the Variational Quantum Algorithm comes from using classical machine learning to find the parameters of the gates that minimize </div><div>the ground state of the Ising Hamiltonian.</div><div><br /></div><div>One thing that is easy for the Adiabatic Quantum Computation to do is quantum error correction. </div><div>The adiabatic theorem tells you that the longer you wait for your system to reach its final state, </div><div>the more likely you are to have stayed in the ground state (or, said another way, the less likely you are to have excitations, </div><div>which basically translate to errors). Therefore, in order to reduce error, all you need to do is run the algorithm in </div><div>the adiabatic quantum computer for a longer time. If you ran it for T = infinity, you'd be 100% accurate.</div><div><br /></div><div><br /></div><div>Mapping a problem to a basic Boolean satisfiability problem can difficult or can create added work in the translation</div><div>(imagine a problem that requires the number of qubits to scale with the number of real variables by a polynomial of high degree, or worse, exponentially). </div><div>You might also find that an approximate solver is not good when you have hard constraints that must not be violated </div><div>(these machines operate at a finite time scale, therefore there is always some noise in the adiabatic quantum computer). Therefore it is key to understand</div><div>how such systems work with a mixture of synchronized and unsynchronized states, to understand our chimera state for a given system in effect. </div><div><br /></div><div><br /></div><div>As discussed, chimera states can form naturally in systems that can be treated as systems of oscillators, like the quantum systems we have discussed. </div><div>This is really the main reason why quantum systems are being considered applicable for solving these kinds of Boolean SAT problems in the first place.</div><div> </div><div>Although, as we have seen in a previous video, the quantum systems can be very sensitive to initial conditions and in effect create chaos </div><div>even in controlled (and isolated) quantum systems, we see from looking at the mixture of asynchronous and synchronous elements in chimera states in such systems as </div><div>in fact beneficial to driving an adiabatic quantum network to achieve quantum error correction but at the same time allowing for it</div><div>to behave as a meta-heuristic state so that we never have the system becoming trapped in solutions, corresponding to local minima, which are not the globally </div><div>optimized solution to the problem. </div><div><br /></div><div>In effect our quantum chimeras really offer the best signature for the quantum system we are interested in using </div><div>for solving the kinds of problems we may want a quantum network to solve.</div><div><br /></div><div><br /></div><div>The system’s response to the presence of a chimera state is a sharp transition at a critical value of a variable p, </div><div>above which percolation occurs but below which it doesn’t occur. </div><div>Near this critical value, the system is very sensitive to minor perturbations, </div><div>and a number of intriguing phenomena (such as the formation of self-similar fractal patterns as seen in our discussion of chaotic systems) </div><div>are found to take place at or near this transition point, which are called critical behaviours or percolation thresholds. </div><div><br /></div><div>Many complex systems, including biological and physical networks, are considered to be utilizing such critical behaviours</div><div>for their self-organizing and information processing purposes. For example, there is a conjecture that animal nervous systems </div><div>tend to dynamically maintain critical states in their neural dynamics in order to maximize their sensitivity responses and information processing </div><div>capabilities. Such self-organized criticality in natural systems has been a fundamental research topic </div><div>in complex systems science and relates intimately to the study of percolation thresholds with these percolation thresholds corresponding to chimera states. </div><div><br /></div><div>Moreover, the link between the percolation dynamics of complex networks has not been lost on those that</div><div>employ statistics in the analysis of such networks, as renormalization group theory, a technique straight from </div><div>the quantum field theory toolkit, is sometimes used to quantify the percolation thresholds of complex systems.</div><div><br /></div><div>The real amazing thing is that so many of the problems that are solvable by such quantum networks, </div><div>in the classical world, should be solvable in the first place.</div><div><br /></div><div>Not taking this amazement for granted, we may have to think about the emergent non-linear behaviour and dynamics of the real world as merely an echo of the balance </div><div>of synchronization and chaos that exist in the quantum mechanical foundations of the whole universe, </div><div>an echo of the quantum chimera that happens to shows up into our reality.</div></div><div><br /></div><h2 style="text-align: left;"><br /></h2>MuonRayhttp://www.blogger.com/profile/03712859045968965104noreply@blogger.com0tag:blogger.com,1999:blog-5996268492156364201.post-51665175434018912952021-04-12T13:30:00.002-07:002021-04-12T13:30:56.137-07:00Drone Lidar Laser Test Flight - Scanning of An Irish Coastline<iframe frameborder="0" height="270" src="https://youtube.com/embed/k7npJXYCuLg" width="480"></iframe><div><br /></div><div><span style="background-color: #f9f9f9; color: #030303; font-family: Roboto, Arial, sans-serif; font-size: 14px; white-space: pre-wrap;">Here I present a summary of a drone-borne Lidar laser flight test that took place in Ireland on December 2020 based on a concept of practical applications of a Drone-Borne Topography and Bathymetric LiDAR Sensor Technology that uses modified, wave-length selective drone cameras in tandem with specially selected lasers.
The laser platform showcased is designed to send laser pulses in 2 circular rings at 2 kHz. The echo signal of each laser signal is digitized and recorded for the entire range gate of 50m. This is the selectable range from which the returning signals can be measured. This range is used to separate echoes from different regions of scattered targets.
The determination of accurate bathymetric information using precision laser scanning technology on LiDAR devices is a key element for near offshore activities, hydrological studies such as coastal engineering applications, sedimentary processes, hydrographic surveying as well as archaeological mapping and biological research. UAV imagery processed with Structure from Motion (SfM) and Multi View Stereo (MVS) techniques can provide a low-cost alternative to established shallow seabed mapping techniques offering as well the important visual information.
Modern photogrammetry and remote sensing have found small Unmanned Aerial Vehicles (UAVs) to be a valuable source of data in various branches of science and industry (e.g., ecosystem monitoring, agriculture, archaeology and construction). Recently, the growing role of laser scanning in the application of UAVs has also been observed in tandem with conventional 3D scanning using imagery.
Drones equipped with specially selected lasers can offer new applications in hydrographic applications such as the production of coastline and inland waterway profiles.
The advent of UAVs as carrier platforms of active and passive mapping sensors had a huge impact in the field of photogrammetry and remote sensing. The introduction of Structure-from-Motion (SfM) and Dense Image Matching (DIM) techniques providing automatic orientation of entire image blocks and height estimates for every image pixel has democratized image-based 3D mapping of topography and dramatically increased the achievable point densities. The applicability of UAV-photogrammetry is further facilitated due to the existence of easy-to-use software solutions providing an automated processing chain from captured images to digital surface models, 3D meshes, and orthophoto maps, respectively.
While the application of SfM is directly applicable for the dry part of alluvial and coastal areas, mapping of underwater topography requires consideration of beam bending at the air-water medium boundary.
Water refraction poses significant challenges on depth determination. This problem has been addressed through customized image-based refraction correction algorithms or by modifying the collinearity equation. It will be the task of image processing to then use a full waveform processing algorithm for analyzing waveforms. Ideally false signals are eliminated and the most probable wave model is determined. Finally, data sets with high accuracy, high resolution and hydrography are provided to support point classification. Machine Learning techniques are constantly being developed to bridge some of the technological gaps in this field.
Technology Developed and Tested by MuonRay Enterprises Ireland.
(Copyright MuonRay Enterprises Ireland)</span></div>MuonRayhttp://www.blogger.com/profile/03712859045968965104noreply@blogger.com0tag:blogger.com,1999:blog-5996268492156364201.post-25200859783208693072021-03-06T18:21:00.005-08:002021-06-14T17:51:52.475-07:00Quantum Chaos Theory: Nature's Emergence of Fractal Complexity Using Quantum Mechanics Across Scales<iframe frameborder="0" height="270" src="https://youtube.com/embed/c8QWWkEmEAg" width="480"></iframe><div><br /></div><div><div><br /></div><div><br /></div><div>One of the puzzling aspects of the different theoretical models in physics is the question of why should the laws of physics behave differently</div><div>at different scales if there should be no absolute frames of reference in the universe as a whole. </div><div><br /></div><div>It sort of the opposite to the argument given in favour of the turtle-stacking model of suspension of the Earth where a person who believes that</div><div>the earth is suspended on the back of a giant turtle and is asked "but what suspends the turtle?" to which he or she answers: "simple. It's turtles all the way down!"</div><div>this example, though humorous, does reflect our need for sensible models to describe non-local behaviour, be it the position of the earth in space or electrons</div><div>in an atom.</div><div><br /></div><div>If we are to accept, with equal but scientifically verifiable passion, how the fundamental particles that make up atoms, molecules and matter are themselves suspended </div><div>to the rules of quantum theory could we not also make a similar argument to turtle stacking, but in reverse? That is to ask: </div><div>is it really quantum all the way from down to up? </div><div><br /></div><div><br /></div><div>The correspondence principle states that classical mechanics is merely the classical limit of quantum mechanics, specifically in the limit as the ratio of </div><div>Planck's constant as the action of the system tends to zero. </div><div><br /></div><div>So in certain interpretations of Quantum theory, such as the path integral interpretation, we see the cancelling out the quantum effects </div><div>by the separate particle histories causing decoherence with one another. </div><div>So that in effect we are expected to assume that we loose some of the quantumness "in the wash" so to speak</div><div>from the microscopic world to the mesoscopic and macroscopic. </div><div><br /></div><div><br /></div><div>The question that still arises however is that if quantum theory suspends the behaviour of atoms, matter and everything else from the ground up then </div><div>shouldn't there be some quantum effects that translate across the scales? Even relatively small quantum effects?</div><div><br /></div><div>In a sense this is the sameness problem in physics, where we can ask why do we not </div><div>see many of the features that govern quantum effects reappear at larger scales in some aspects of structure?</div><div><br /></div><div>Unlike the turtle stacking analogy, this is not an unreasonable question to ask, after all as quantum effects occur at small scales so their must be a lot of </div><div>these effects that add up even in an ordinary piece of matter. Macroscopic Magnetic effects for example is the amplification of the </div><div>individual magnetic moments of atoms in a a material which are related directly to the quantum spin of individual atoms and electrons.</div><div><br /></div><div>If there is not a sameness across all physical theories then we may be forced to admit that many of the separate subsets of physics are not branches as much as they </div><div>are disjointed appendages assumed to be glued together by some insofar unwitnessed "unification". </div><div><br /></div><div>It should be recognized that just because we have a set of theories that are able to grind up </div><div>experimental data and churn out predictions does not mean we truly understand the principles behind the theory and we may be just working with </div><div>the mathematical physicists equivalent of a black box that only lets us witness the inputs and outputs in a blinding "shut-up and calculate" fashion.</div><div> </div><div> </div><div>In Mathematics, the application of Fractals in geometry provides a clear insight to this concept of sameness being apparent across scales, </div><div>being made popular by the mathematics of Gaston Julia, Benbot Mandelbrot and others. </div><div><br /></div><div>In Fractal Geometry we see that complex geometical patterns, some of which even begin to imitate the kind of patterns we see in the natural world, have </div><div>this principle of sameness of general shape operating at different scales. </div><div><br /></div><div>What is most surprising is how these patterns can emerge from simple rules of continuous </div><div>iteration, without the need for specific coded instructions to create the precise shapes of the patterns. </div><div>In bifurcation diagrams for example we see patterns emerge from a combination of following a computation with a randomly varying term or</div><div>set of terms added to it. The potentially infinite, often repeating patterns could not have been explicitly coded, as infinite amounts of instructions</div><div>would be required to be translated in an arcane fashion into such code. They arise from the balance between</div><div>exploration and exploitation in the system: the core feature of a meta-heuristic procedure.</div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div><br /></div><div>We can also begin to see that unlike the relatively abstract and idealized geometry we are forced to learn in school about perfect cubes, spheres, cones and so on, </div><div>this fractal geometry seems to create the kind of shapes and patterns that are seen in the physical phenomena of the real world- those of the </div><div>shapes of mountains, coastlines, river systems, blood vessels, clouds, continents and even the vast networks of galaxy clusters as seen in the large scale universe. </div><div><br /></div><div>Shapes that are visually invariant across scales. </div><div><br /></div><div>It becomes apparent when studying the mathematics of scalar fields and how they couple that we can see kinds of physical power law systems emerging </div><div>from networks of scalar fields that are coupled with one another and how similar this appears to the mathematics of discontinous pas-coupling, </div><div>a metaheuristic technique that involves a defined signalling term with a randomly oscillating delay term to achieve an emergent equilibrium.</div><div>In effect we see a duality between such scalar fields and self-synchronizing quantum networks.</div><div><br /></div><div>The emergent synchronization and equilibrium of these systems is favored as being the energetic ground sate of the system, that the state evolves toward over time.</div><div><br /></div><div>Even in systems that are not explicitly programmed to achieve this kind of emergent adaptive network, such as naturally occurring quantum systems</div><div>(for example nanoribbons, bose-einstein condensates, networks of quantum spins in magnetic materials to name a few) it can arise naturally by </div><div>simply having the individual nodes exploit a power law for coupling with a randomly varying term.</div><div><br /></div><div>As we discussed in a previous video, many of the fundamanetals of quantum theory behave this way, exploiting a very precise and simple series of </div><div>momentum exchange rules with a randomly varying probabilistic term to create the kind of emergent behaviour that we see in so-called pure quantum systems.</div><div><br /></div><div> </div><div>If this is true, then there must be quantum mechanisms underlying classical chaos in such systems.</div><div><br /></div><div>In our previously discussed model of the 2D Quantum Newton’s cradle, the balls are replaced by our signalling atoms or electrons, confined in rows.</div><div>Adding additional momentum, such as a photon from a laser, can kick the atoms into motion, causing them to oscillate back and forth just as in </div><div>the classical Newton's Cradle. </div><div><br /></div><div>However, unlike the toy, the atoms in a quantum newtons cradle can both collide and pass through one another because of the oddities of quantum physics, such as </div><div>quantum tunneling.</div><div>This leads to a sum over histotires of the different paths a particle can travel. </div><div><br /></div><div><br /></div><div>Just as in classical mechanics, With our quantum Newton's Cradle as the strength of the interaction increased, the motion of each cradle’s atoms in the arrangement</div><div>can transitioned from periodic to chaotic.</div><div><br /></div><div>This is equivalent of the the momentum space distribution of the atoms approaching a thermal distribution over a frequency of time</div><div>indicating the system is reaching some equilibirium. In effect we have a synchronized system of atom to responds collectively.</div><div><br /></div><div>In certain quantum transition effects, with thermalisation of groups of atoms, we also see the effects of quantum chaos which create</div><div>complex structure from simple momentum transition rules. A small section of electrons in the thermalised system, when perturbed, can cause</div><div>interactions which have effects that iterate out into the entire system even without direct contact between individual electrons. non-local behaviour in effect.</div><div><br /></div><div>the spectral properties of non-interacting two-dimensional electrons in a magnetic field in a lattice can also create self-similar fractals</div><div>first discovered in the 1976 Ph.D. work of Douglas Hofstadter. Hofstadter described the structure in 1976 in his modelling of the energy levels </div><div>of Bloch electrons in magnetic fields.[1] It gives a graphical representation of the spectrum of Harper's equation at different frequencies. </div><div>The intricate mathematical structure of this spectrum was independently discovered by Soviet physicist Mark Azbel in 1964 (the Azbel-Hofstadter model),</div><div>[4] but Azbel did not plot the structure as a geometrical object.</div><div><br /></div><div><br /></div><div>The fact is that the non-linear effects of imperfections and random behavior in additional to the quantum exchanges during photon-electron interactions </div><div>and electron-electron interactions will inevitably lead to the same kind of emergent chaos that is seen in classical systems such that a small change </div><div>in the position of an atom arrangement in a crystal lattice or the random excitation of a quasiparticles momentum state will inevitably lead to </div><div>an emergent pattern which will be drastically different to the original pattern</div><div><br /></div><div><br /></div><div>by creating a grid of 2D atoms we can simulate this effect and make a relationship between the system behaviour across the map of the fractal pattern.</div><div><br /></div><div>The grid is the energy surface the atoms are binded to. Mathematically this is a matrix, the hamiltonian.</div><div>It is a general principle of Quantum Mechanics that there is an operator for every physical observable, for energy and momentum operators </div><div>for example, which can be measured.</div><div>In a system that is defined by a wavefunction, which is an eigenfunction, acts on an operator</div><div>then the system is said to be in an eigenstate. The values for energy or momentum operators are therefore eigenvalues.</div><div><br /></div><div>In the 2d square grid we can represent the evolution of the eigenstate as an emergent fractal pattern with the pattern being highly ordered</div><div>and dependent on the slight tweaking of the initiral conditions of the atoms topology in the lattice.</div><div><br /></div><div>for example lets take the idea of 2d electrons in a square lattice and compare with a hexagonal lattice</div><div><br /></div><div><br /></div><div>In The arrangement of atoms The momentum exchange rules are the same in each case however due to the position of the atoms creating changes in </div><div>the small and randomised position of the particles, that emergent pattern will be completely different.</div><div><br /></div><div><br /></div><div>Even though our change in arrangement was simple, the emergent fractal nature of the spectrum shows completely different results.</div><div>this is not decoherence of any kind, the system is still behaving as an isolated thermal bath. However a local variation causes perturbations that </div><div>reverberate throughout the network in a complex and adaptive system that reinforces itself. </div><div><br /></div><div>The emergent nature of the different energy level structure is also apparent, with the electron transition regime in the hexagonal lattice</div><div>now appearing much more relativistic as compared to the 2D lattice.</div><div><br /></div><div><br /></div><div>the onset of chaotic behavior in the system can be be used to describe how interacting quantum particles drive certain materials,such as graphene or </div><div>superconducting crystals to a thermal equilibrium. </div><div><br /></div><div>This insight is important to note as many quantum technological devices are being considered that rely on nonequilibrium quantum effects. Of particular interest are </div><div>devices that use the cuprate high temperature superconductors, most notably BSCCO which has a crystal structure that behaves as</div><div>a natural form of josephson junction. These are considered as one of the most promising elements in quantum sensors and as potential processors in </div><div>the much touted field of neuromorphic quantum computing, using scalar coupling that can occur between separate josephson junctions in quantum circuits.</div><div><br /></div><div>However many of the design considerations of both quantum-based sensors and as hardware for quantum computing seem to ignore the effects of quantum chaos, </div><div>or else think that the description of quantum chaos is somehow not important to the development of their "machines" quote un quote.</div><div><br /></div><div>Fundamentally it is still an unsettling definition as to why quantum mechanical systems is framed to be in the domain of what we arbitrarily call "small" </div><div>especially when we see effects in the laboratory, such as quantum entanglement, superconductivity, superfluidity and bose-einstein condensation, </div><div>that have nothing to do with small length or time scales.</div><div><br /></div><div>So the foundations of quantum theory may not really be dependent on what we refer to as the size scale the physical laws operate upon but as a more underlying factor</div><div>that operates independently across different scales, the factor of an emergent quantum chaos that goes on to define a quantum systems behavior.</div><div><br /></div><div>A quantum sameness that is conceptually just as impressive , and may even be complimentary to, the concept of fractal sameness across scales.</div><div><br /></div></div>MuonRayhttp://www.blogger.com/profile/03712859045968965104noreply@blogger.com0tag:blogger.com,1999:blog-5996268492156364201.post-26642173732281274292021-02-22T15:27:00.000-08:002021-02-22T15:27:08.625-08:00Heuristics, Quantum Computers and Artificial Intelligence<div dir="ltr" style="text-align: left;" trbidi="on">
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Introduction</h2>
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The term computer has had many different incarnations of meaning over the past 100 years. From the human computers at Los Alamos, the room-sized valve computers to modern semiconductor computers, the word computer had become far more generalized than its original definition.<br />
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The most universal definition of a computer I can think of is this:<br />
A computer is a logic filing system functioning within the parameters of a particular hardware.<br />
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Modern computers use programs, themselves merely sophisticated filing systems, functioning in the parameters of the Boolean logic of computer hardware, which is a series of semiconductor chips and electrical components.<br />
The apparent complexity of computation is, in reality, sending and retrieving a series of previously written instructions. Computer programs need to execute functions by calling on a series of header files written in a particular language to work through a continuum of instructions to perform any task. these header files are themselves made up of instructions, written in a computer language, which itself is a substitution of a more fundamental and arcane machine code, ultimately in binary 0s and 1s.<br />
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Computation as we know it, i.e. a Turing Machine, must use this method if it is constructed with switching elements be they valves or transistors.<br />
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Some problems which Turing Machines are used to solve are nondeterministic, i.e. a program that is, or depends on, a random number generator running in polynomial or exponential timescales. these timescales are such that they are equal to or grater than the number of operations (arithmetic, sorting, matchings, ect).<br />
The operations of this class, known as P operations (P for Polynomial time) are the most tractable problems in complexity theory and are at the foundation of all complex computations. Due to them being deterministic it is possible for a computer to know how long it takes to solve these problems and so it is no surprise that most functions of a computer use algorithms that break down most hard problems into P operations, such as the case of the halting problem which defines the Turing-complete model of computation.<br />
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Due to the use of a a random number generator, solutions to the algorithm will be nondeterministic at such timescales. These are NP operations. Although any given solution to such a problem can be verified quickly by approximation or by randomization (as in Monte Carlo simulations) there is no known efficient way using a computer to locate a solution in the first place. The most notable characteristic of NP-complete problems (where every element is NP) is that no fast solution to them is known because the timescales involved are completely random. NP-complete problems could take anywhere between 10 seconds and 10 trillion years to solve, even if they are relatively simple.<br />
However, the solution becomes deterministic if checked by a deterministic Turing subroutine that solves a different problem that uses polynomial time excluding the time within the subroutine. Intuitively, a polynomial-time reduction proves that the first problem is no more difficult than the second one, because whenever an efficient algorithm exists for second problem, one exists for the first problem as well. These are NP-hard problems.<br />
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Assuming not all NP problems can be solved withing polynomial time, i.e <b style="background-color: white; font-family: sans-serif; font-size: 12.7273px; line-height: 19.2px;">P</b><span face="sans-serif" style="background-color: white; font-size: 12.7273px; line-height: 19.2px;"> ≠ </span><b style="background-color: white; font-family: sans-serif; font-size: 12.7273px; line-height: 19.2px;">NP, </b>the measure of complexity of such problems can be graphed in this fashion:<br />
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Most NP-Complete and NP-hard problems are optimization and search problems, such as the travelling salesman problem or Langton's ant, Langton's Loops, Turmites and Turing-type cellular automatons in general.<br />
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The K-SAT problem plays a most important role in the Theory of Computation (NPcompleteproblem) The setup of the KSAT Computation (NP complete problem). The setup of the K-SAT problem is as follows<br />
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• Let B={x1, x2, …, xn} be a set of Boolean variables.<br />
•Let Cibe a disjunction of k elements of B<br />
•Finally, let Fbe a conjunction of m clauses Ci.<br />
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Question: Is there an assignment of Boolean variables in<br />
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F=1?<br />
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More advanced versions of these algorithms require large timescales to work with, which is not much use for search results for the here and now. Therefore we sometimes need the computational equivalent of the educated guess. It turns out there is such a thing. They are called heuristics.<br />
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(<b>note: we do not yet know if </b><b style="background-color: white; font-family: sans-serif; font-size: 12.7273px; line-height: 19.2px;">P</b><span face="sans-serif" style="background-color: white; font-size: 12.7273px; line-height: 19.2px;"> ≠ </span><b style="background-color: white; font-family: sans-serif; font-size: 12.7273px; line-height: 19.2px;">NP, i.e. if there are NP-problems that are harder to compute than to verify in polynomial time, or if P = NP, i.e. all NP-problems can be solved and verified in polynomial timescales. It is highly suspected that </b><b style="background-color: white; font-family: sans-serif; font-size: 12.7273px; line-height: 19.2px;">P</b><span face="sans-serif" style="background-color: white; font-size: 12.7273px; line-height: 19.2px;"> ≠ </span><b style="background-color: white; font-family: sans-serif; font-size: 12.7273px; line-height: 19.2px;">NP, or at least that all NP problems may be tractable by heuristics, but there may be no way to prove any of these conjectures</b><br />
<b style="background-color: white; font-family: sans-serif; font-size: 12.7273px; line-height: 19.2px;">If a person or team finds a proof, the Clay Mathematics Institute will award the winner with $1 Million to the first correct solution)</b><br />
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Heuristics and Metaheuristics.</h2>
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The word heuristic comes from the greek verb "<i style="background-color: white; font-family: sans-serif; font-size: 12.7273px; line-height: 19.2px;">heuriskō" </i><span face="sans-serif" style="background-color: white; font-size: 12.7273px; line-height: 19.2px;">which translates to</span><i style="background-color: white; font-family: sans-serif; font-size: 12.7273px; line-height: 19.2px;"> "I find". </i><span face="sans-serif" style="background-color: white; font-size: 12.7273px; line-height: 19.2px;">A related term is the famous interjection</span><i style="background-color: white; font-family: sans-serif; font-size: 12.7273px; line-height: 19.2px;"> "Eureka!" </i><span face="sans-serif" style="background-color: white; font-size: 12.7273px; line-height: 19.2px;">or </span><i style="background-color: white; font-family: sans-serif; font-size: 12.7273px; line-height: 19.2px;">"</i><i style="background-color: white; font-family: sans-serif; font-size: 12.7273px; line-height: 19.2px;">heúrēka", </i><span face="sans-serif" style="background-color: white; font-size: 12.7273px; line-height: 19.2px;">which was said to be uttered by Archimedes when he solved the problem of finding volume of irregular solids.</span></div>
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In computer science, a heuristic is an algorithm that solves a particular problem, but may not give the optimum solution in all cases. In some cases, a heuristic is used because a given problem cannot be solved optimally in every case in a reasonable amount of time or space. For example, progamming language compilers use many different algorithms to generate assembly code from the source language (C/C++/Java/etc). In many cases, these algorithms are heuristics. Heuristics are useful for some problems that a compiler must address which belong to a class of problems known as NP Problems, which are computationally difficult to solve. </div>
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In other cases, a heuristic is used because the compiler does not or cannot have enough information to make the best decision. Compiler writers attempt to create heuristics that solve the given problems well in most cases, and in a reasonable amount of time or space, and/or with limited information.</div>
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A heuristic is a low level function of a compiler program and is essentially a simple strategy which processes the computational capacity in order to optimize it or to instruct either the computer or programmer what to do if the program is outside the limits of some boundary.<br />
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If you have ever written a computer program on any compiler , in C, C++, Matlab, Fortran, ect, when you compile the program a heuristic essentially searches the program when it is compiling to try and optimize it or alert the computer to shut it off if the program is beyond computational capacity or outside the limits of some function you call to run the program.<br />
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Anyone who does programming knows that the messages that are given by the computer by this heuristic are terrible due the limited information available, making them vague. In most cases they function as warnings and rely on the intelligence of the programmer to solve the problem.<br />
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An error in a C++ and Matlab program, as shown by a heuristic<br />
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Some Examples of compiler heuristics include:</div>
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<li style="margin-bottom: 0.1em;">Inlining decisions</li>
<li style="margin-bottom: 0.1em;">Unrolling decisions</li>
<li style="margin-bottom: 0.1em;">Packed-data (SIMD) optimization decisions</li>
<li style="margin-bottom: 0.1em;">Instruction selection</li>
<li style="margin-bottom: 0.1em;">Register allocation</li>
<li style="margin-bottom: 0.1em;">Instruction scheduling</li>
<li style="margin-bottom: 0.1em;">Software pipelining</li>
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The problem with heuristics is that no assumptions are made about what parts of the program can be skipped or cut away, any suggestions or tip must always be programmed in from previous experience by the user but the heuristic itself does not display machine learning. Even if the computer had done the same problem a thousand times the heuristic will not learn to navigate through a program, searching for the best route to perform the computation. In cases where the software is small, a single programmer can solve the problem easily but when the software contains millions of lines of code it becomes a problem which even the biggest computer industries struggle with.<br />
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Since heuristics are encoded strategies, they can be generalized into ways to solve any problem by using approximations. By using approximations, speed can be increased but accuracy is sacrificed.<br />
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An example of a problem which could be solved by a heuristic is the <span face="sans-serif" style="background-color: white; font-size: 12.7273px; line-height: 19.1903px;">problem of finding the shortest path between two vertices (or nodes), or intersections, on a network graph or road map. The goal is then to have a result </span><span face="sans-serif" style="background-color: white; font-size: 12.7273px; line-height: 19.1903px;">such that the sum of the </span><a href="http://en.wikipedia.org/wiki/Glossary_of_graph_theory#Weighted_graphs_and_networks" style="background-color: white; background-image: none; color: #0b0080; font-family: sans-serif; font-size: 12.7273px; line-height: 19.1903px; text-decoration: none;" title="Glossary of graph theory">l</a>engths<span style="background-color: white;"><span face="sans-serif"><span style="font-size: 12.7273px; line-height: 19.1903px;"> of its constituent vertices, or road segments, is minimized. This can be </span></span><span face="sans-serif" style="font-size: x-small;"><span style="line-height: 19.1875px;">generalized</span></span><span face="sans-serif"><span style="font-size: 12.7273px; line-height: 19.1903px;"> to any given geometry from which connection probabilities could be estimated.</span></span></span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiF9iAI0JVaPvClR6_hqq9M2T4dbFvwUkhBq1s1lvYTSu6gTDmAr8_1Cxn9ckLXJ2jBRbkx9otpxKh5giPFgz9eZWV6o-7_5Pa769bwA6uPNTEFHHVQPADe8UTpY42oLoMspMPUV0gXB-lC/s1600/Shotest+Path.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="242" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiF9iAI0JVaPvClR6_hqq9M2T4dbFvwUkhBq1s1lvYTSu6gTDmAr8_1Cxn9ckLXJ2jBRbkx9otpxKh5giPFgz9eZWV6o-7_5Pa769bwA6uPNTEFHHVQPADe8UTpY42oLoMspMPUV0gXB-lC/s400/Shotest+Path.png" width="400" /></a></div>
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<span face="sans-serif" style="background-color: white; font-size: 12.7273px; line-height: 19.1903px;">We could say a high-value heuristic for this problem is one which computes a path quickly, perhaps ignoring the underlying geometry and merely weighting the network topology. However by doing this the path computed might not be really be the shortest. A low-value heuristic computes a path more slowly, perhaps focused on the topology and underlying geometry where the path chosen as the solution becomes shorter. </span><br />
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<span face="sans-serif" style="background-color: white; font-size: 12.7273px; line-height: 19.1903px;">A computer using such heuristics however would not never know which heuristic to compute to solve a particular problem; it could use a high value heuristic for a small problem and a low value heuristic for a large problem which would could negate the point in using a heuristic at all. In this sense a normal computer heuristic functions just the same as the </span><span face="sans-serif" style="font-size: x-small;"><span style="line-height: 19.1875px;">Turing subroutine in NP-hard problems, in that it tries and solves the problem by solving a different problem that uses polynomial time excluding the time within the subroutine.</span></span><br />
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<span style="background-color: white;"><span face="sans-serif"><span style="font-size: 12.7273px; line-height: 19.1903px;">In many cases this is true and heuristics are merely used to facilitate a human-computer interface. By applying fuzzy logic to complex data systems, programmers have found that it helps facilitate human intuition in the experience by creating near-solutions to problems. The best example of which is Internet search engines, which are NP-hard problems, such as Google which search through truly massive datasets and create a series of near-matches to often vague search criterion, allowing human intuition to take over on interpreting it. </span></span></span><br />
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<span style="background-color: white;"><span face="sans-serif"><span style="font-size: 12.7273px; line-height: 19.1903px;">This makes Google search popular, it is a simpler and a more instinctive way to search for something and is not bound by being overly specific on search criteria. So in this case, using fast but only </span></span><span face="sans-serif" style="font-size: x-small;"><span style="line-height: 19.1875px;">moderately</span></span><span face="sans-serif"><span style="font-size: 12.7273px; line-height: 19.1903px;"> specific heuristics does have a benefit for human use of computers, making it more intuitive to use something as complex as a complete search of something as large as the internet.</span></span></span><br />
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<span style="background-color: white;"><span face="sans-serif"><span style="font-size: 12.7273px; line-height: 19.1903px;">So, If we want to be direct about it, a heuristic does not really help a computer solve fundamental problems as the normal constraints of problem solving, namely speed versus accuracy, are always being balanced. To make heuristics more </span></span><span face="sans-serif" style="font-size: x-small;"><span style="line-height: 19.1875px;">useful</span></span><span face="sans-serif"><span style="font-size: 12.7273px; line-height: 19.1903px;"> in areas beyond computer-human interfaces and into the realms of artificial intelligence we need to see if we can </span><span style="font-size: x-small;"><span style="line-height: 19.1875px;">extend what a</span></span><span style="font-size: 12.7273px; line-height: 19.1903px;"> heuristic can be</span></span></span><span face="sans-serif" style="background-color: white; font-size: 12.7273px; line-height: 19.1903px;">.</span><br />
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The most obvious way to advance heuristics might be seeing what other applications can we squeeze out of current heuristics. However, it is often better to look at what heuristics can't do if we want to improve them.<br />
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As discussed heuristics suffer from 3 main drawbacks:<br />
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(1) Heuristics do not make assumptions on the task being solved are made.<br />
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(2) Heuristics do not always make globally optimized solutions.<br />
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(3) Heuristics themselves at a given task are not rated as being good or bad relative to other heuristics at a different task.<br />
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The first drawback could be remedied, as Google have done, by brute force paralell processing to create "did you mean x,yor z, instead of a,b or c?" in search criteria. Indeed this technique is also used in Google's language translation heuristics. Methods of language translation, using experimental methods beyond brute force heuristics are an area of active research and are non-trivial problems for computers.<br />
By simply readdressing the problem by generating a question heuristic this makes the search algorithm a metaheuristic, as a higher level heuristic is summoning the function of a lower one. Asking to repeat the question but also to perform the search assumes that the solution is not a global optimum solution so the intrinsic nature of heuristics is still present.<br />
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By creating machine learning, it is possible to modifiy these metaheuristics using neural networks which would process the heuristics themselves to try and weight the most likely paths based on multiple variables.<br />
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These neural network programs can be used as a form of artificial intelligence (AI) and have spawned millions of intricately constructed computer programs which have a wide variety of uses. Some of their uses is in signal analysis of large amounts of datasets with many variables, which can be used in large experiments such as at the LHC at CERN and NASA's Kepler Mission.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjOIsMi130l4SBQTECFrwLt0XQhI1bYpfLY3N626V6EyQQmqucpVjKWGCqfGTz_jA1-RjpeXohv8M8lKkLEwNJQR2YDX1r-f8VU8nLEdumYRs4Ja7wCsN0f93neZ3012gGGU0vHPKYNaYug/s1600/Higgs.gif" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="198" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjOIsMi130l4SBQTECFrwLt0XQhI1bYpfLY3N626V6EyQQmqucpVjKWGCqfGTz_jA1-RjpeXohv8M8lKkLEwNJQR2YDX1r-f8VU8nLEdumYRs4Ja7wCsN0f93neZ3012gGGU0vHPKYNaYug/s400/Higgs.gif" width="400" /></a><br />
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<i style="font-family: inherit;">Multilayer Perceptron Neural Network assisted reconstructed signal of a Higgs Boson decay event, isolated from background effects.</i></div><div><i>(performed in CERN ROOT Code)<br /></i>
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<a href="http://www.scientificamerican.com/sciam/cache/file/79ADD2B6-3B32-494C-BA7AA3D18B680DA9_article.jpg?8E597" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img alt="Kepler spacecraft field of view" border="0" src="http://www.scientificamerican.com/sciam/cache/file/79ADD2B6-3B32-494C-BA7AA3D18B680DA9_article.jpg?8E597" /></a></div>
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<span style="font-family: inherit;"><i><span style="line-height: 24px;">Field of view of Kepler spacecraft, trained on a patch of more than 150,000 stars near the constellation Cygnus, trails Earth as both orbit the sun. </span><span style="line-height: 24px;">One common false positive Kepler is trained to notice is </span><span style="color: black;">an eclipsing binary star<span style="line-height: 24px;"> behind one of Kepler's target stars; the background light blocked by such an eclipse can mimic the periodic dimming that Kepler uses to identify planets passing in front of its target stars.</span></span></i></span></div>
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More generally, neural networks can be used in full image reconstructions and shape recognition as well as in error analysis. These programs have some level of self correction and can learn in a sense. However they are not self-sufficient as "training" programs must be fed to the program before it can be at a level where it can correct itself.<br />
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Such A.I. metaheuristics may resemble global optimization problems similar to classical problems such as the travelling salesman, ant colony or swarm optimization, which can navigate through maze-like databases. The solutions, although not completely optimal, are far more useful than a typical heuristic would be. If used in parallel processing, such heuristics display machine learning which do make assumptions on the task being solved. This means that error analysis is possible with these heuristics.<br />
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Metaheuristics are also used in Monte Carlo method simulations to create global optimization, which are widely used in physics and science in general to solve problems. One such problem is Simulated Annealing.<br />
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Simulated annealing copies a phenomenon in nature--the annealing of solids--to optimize a complex system. Annealing refers to heating a solid and then cooling it slowly. Atoms then assume a nearly globally minimum energy state. In 1953 Metropolis created an algorithm to simulate the annealing process. The algorithm simulates a small random displacement of an atom that results in a change in energy. If the change in energy is negative, the energy state of the new configuration is lower and the new configuration is accepted. If the change in energy is positive, the new configuration has a higher energy state; however, it may still be accepted according to the Boltzmann probability factor:</div>
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where is the Boltzmann constant and T is the current temperature. By examining this equation we should note two things: the probability is proportional to temperature--as the solid cools, the probability gets smaller; and inversely proportional to --as the change in energy is larger the probability of accepting the change gets smaller.</div>
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When applied to engineering design, an analogy is made between energy and the objective function. The design is started at a high “temperature”, where it has a high objective (we assume we are minimizing). Random perturbations are then made to the design. If the objective is lower, the new design is made the current design; if it is higher, it may still be accepted according the probability given by the Boltzmann factor. The Boltzmann probability is compared to a random number drawn from a uniform distribution between 0 and 1; if the random number is smaller than the Boltzmann probability, the configuration is accepted. This allows the algorithm to escape local minima.</div>
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As the temperature is gradually lowered, the probability that a worse design is accepted becomes smaller. Typically at high temperatures the gross structure of the design emerges which is then refined at lower temperatures.</div>
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Although it can be used for continuous problems, simulated annealing is especially effective when applied to combinatorial or discrete problems. Although the algorithm is not guaranteed to find the best optimum, it will often find near optimum designs with many fewer design evaluations than other algorithms. (It can still be computationally expensive, however.) It is also an easy algorithm to implement.</div>
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The Metropolis-Hastings Algorithm generates sample states of a given thermodynamic system.<br />
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The Metropolis–Hastings algorithm can draw samples from any probability distribution <i>P(x)</i>, which can be from the Boltzman Distribution, Fermi-Dirac Distribution or Bose-Einstein Distribution just provided you can compute the value of a function <i>Q(x)</i> which is <i>proportional</i> to the density of <i>P</i>. </div>
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The lax requirement that <i>Q(x)</i> should be merely proportional to the density, rather than exactly equal to it, makes the Metropolis–Hastings algorithm particularly useful, because calculating the necessary normalization factor is often extremely difficult using Bayesian Probability Statistics.</div>
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The Metropolis–Hastings algorithm works by generating a sequence of sample values in such a way that, as more and more sample values are produced, the distribution of values more closely approximates the desired distribution, <i>P(x)</i>. </div>
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<span style="font-size: 12.7273px; line-height: 19.1903px;">These sample values are produced iteratively, with the distribution of the next sample being dependent only on the current sample value (thus making the sequence of samples into a </span>Markov chain<span style="font-size: 12.7273px; line-height: 19.1903px;">). Specifically, at each iteration, the algorithm picks a candidate for the next sample value based on the current sample value. Then, with some probability, the candidate is either accepted (in which case the candidate value is used in the next iteration) or rejected (in which case the candidate value is discarded, and current value is reused in the next iteration)−the probability of acceptance is determined by comparing the likelihoods of the current and candidate sample values with respect to the desired distribution </span><i style="font-size: 12.7273px; line-height: 19.1903px;">P(x)</i><span style="font-size: 12.7273px; line-height: 19.1903px;">.</span></div>
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<span>Simulated Annealing in terms of Metaheuristics</span></h2>
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Simulated Annealing is a stochastic method for optimization problems but can also be considered as a metaheuristic search method for the approximate solution of optimization problems. A typical example is the problem of finding a sequence of different processing steps of a production, so that the orders as quickly as possible without each block to the machine can be carried out. If we denote the set of all possible order with <i>S</i> , then a <i>* x</i> from <i>S</i> to find with minimal execution time <i>c (x *)</i> . If you imagine the cost function <i>c: S -> R</i> as a landscape before, in the high values <i>c (x)</i>a raised dot at the point <i>x</i> mean, it is therefore the aim of finding a deep valley as possible.</div>
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<img align="LEFT" src="http://www.iasor.tu-clausthal.de/Arbeitsgruppen/Stochastische-Optimierung/forschung/Bilder/Stochopt/SimAnnIdee_einf.jpg" width="450" />Starting from a random starting point <i>x0</i> examined simulated annealing random solutions <i>Y</i> in the neighborhood of the current solution <i>x</i> . Y is more than <i>x</i> , that is, applies <i>c (y) <C (x)</i> , then <i>y is </i>accepted as the new solution. In the cost landscape of a downhill path is of <i>x</i> by <i>y</i> taken. Applies the other hand <i>c (y)> c (x)</i> , the path leads from <i>x</i> by <i>y</i> that is uphill, the<i>y</i> still accepted as a new solution - but only with a certain probability, called the acceptance probability. This ability to accept deteriorations, the process may leave local valleys and advance to the global minimum, as indicated in the following figure.</div>
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As acceptance probability for a candidate <i>y</i> at current solution <i>x</i> is usually <i>(- (c (y)-c (x)) / tn) exp</i> is selected. It is <i>tn</i> the so-called temperature with increasing step number <i>n</i> to <i>0</i> goes. Minor deterioration <i>D = c (y)-c (x)> 0</i> are thus more acceptable than larger, even with increasing <i>n,</i> thus decreasing temperature degradations are rarely accepted. If a solution <i>y</i> not accepted, a new candidate is <i>y '</i> from the neighborhood randomly selected.</div>
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<img align="RIGHT" name="graphics1" src="http://www.iasor.tu-clausthal.de/Arbeitsgruppen/Stochastische-Optimierung/forschung/Bilder/Stochopt/law21_AALog_00x.jpg" width="450" />The method is based on a physical model for cooling processes ('annealing') in molten metals, which depends on the temperature control, the material can build up especially favorable structures during solidification. When initially high temperatures many deteriorations are accepted, the process moves usually quickly from the starting point off in the long run, the threshold for acceptance of deterioration is higher, the process comes to rest.</div>
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The order of visited solutions forms an inhomogeneous Markoffkette and it can conditions for the convergence to an optimal solution are given. </div>
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We can go further with these simulation and imagine particles which could operate under the principles of quantum mechanics in such algorithms.<br />
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Metaheuristics and Quantum Computing</h2>
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Quantum computers use entangled particles as qubits. In superconducting Jospehson junctions, the electrons form Cooper Pairs which can undergo quantum tunneling through the superconducting material and the normal conducting or insulating material.<br />
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By thinking about quantum computation in the same way as we think about classical probabilistic computation then we can make active progress in discovering how we can use a quantum computer and how we can use it to perform algorithms such as simulated annealing.<br />
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<span face="sans-serif" style="background-color: white; font-size: 13px; line-height: 19.0469px;">Quantum theory is the language which describes how systems, on the order of Planck's Constant, evolve in time. Thus we are led, in our most fundamental physical theories, to descriptions of systems which are quantum mechanical. These systems, like everything else, can be either analog or digital</span><br />
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A digital system is a system whose configurations are a finite set. In other words a digital system has configurations which you might label as zero, one, two, etc, up to some final number. Now of course, most digital systems are abstractions. For example the digital information represented by voltages in your computer is digital in the sense that you define a certain voltage range to be a zero and another range to be one. Thus even though the underlying system may not be digital, from the perspective of how you use the computer, being digital is a good approximation. </div>
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Analog systems, as opposed to digital systems, are the ones whose configurations aren't drawn from a finite set. Thus, for example, in classical physics, an analog system might be the location of a bug on a line segment. To properly describe where the bug is we need to use the infinite set of real numbers.</div>
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Digital and analog are words which we use to describe the configurations of a physical system (physicists would call these degrees of freedom.) But often a physical system, while it may be digital or analog, also has uncertainty associated with it. </div>
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Thus it is also useful to also make a distinction between systems which are deterministic and those which are probabilistic. Now technically these two ideas really refer to how a systems configurations change with time. But they also influence how we describe the system at a given time. </div>
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A deterministic system is one in which the configuration at some time is completely determined by the configuration at some previous time. A probabilistic system is one in which the configurations change and, either because we are ignorant of information or for some more fundamental reason, these changes are not known with certainty. For systems which evolve deterministically, we can write down that we know the state of the system. For systems which evolve probabilistically, we aren't allowed to describe our system in such precise terms, but must associate probabilities to be in particular configurations.</div>
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<span face="sans-serif" style="background-color: white; font-size: 13px; line-height: 19.0469px;">There is often a confusion between analog systems and probabilistic systems. The first refers to degrees of freedom or configurations. The second refers to how a system changes in time, or what we can predict about it once we make a measurement on it.</span><br />
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Let's consider a single bit. In probability theory, we can describe a bit as having a probability p of being 0, and a probability 1-p of being 1. But if we switch from the 1-norm to the 2-norm, now we no longer want two numbers that sum to 1, we want two numbers whose <em>squares</em> sum to 1. (I'm assuming we're still talking about real numbers.) In other words, we now want a vector (α,β) where α<sup>2</sup> + β<sup>2</sup> = 1. Of course, the set of <em>all</em> such vectors forms a circle:<br />
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The theory we're inventing will <em>somehow</em> have to connect to observation. So, suppose we have a bit that's described by this vector (α,β). Then we'll need to specify what happens if we <em>look</em> at the bit. Well, since it <em>is</em> a bit, we should see either 0 or 1! Furthermore, the probability of seeing 0 and the probability of seeing 1 had better add up to 1. Now, starting from the vector (α,β), how can we get two numbers that add up to 1? Simple: we can let α<sup>2</sup> be the probability of a 0 outcome, and let β<sup>2</sup> be the probability of a 1 outcome.<br />
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But in that case, why not forget about α and β, and just describe the bit <em>directly</em> in terms of probabilities? Ahhhhh. The difference comes in how the vector changes when we apply an operation to it. In probability theory, if we have a bit that's represented by the vector (p,1-p), then we can represent any operation on the bit by a <em>stochastic matrix</em>: that is, a matrix of nonnegative real numbers where every column adds up to 1. So for example, the "bit flip" operation -- which changes the probability of a 1 outcome from p to 1-p -- can be represented as follows:<br />
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<img alt="" src="http://www.scottaaronson.com/cgi-bin/mimetex.cgi?\left(%20\begin{array}0%20&%201\\1%20&%200\end{array}%20\right)\left(\begin{array}%20p\\1-p\end{array}\right)=\left(\begin{array}1-p\\p\end{array}\right)" /></center>
Indeed, it turns out that a stochastic matrix is the <em>most general</em> sort of matrix that always maps a probability vector to another probability vector.<br />
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<span face="sans-serif" style="background-color: white; font-size: 13px; line-height: 19.0469px;">It turns out that the set made of deterministic and probabilistic really has another member, and this member is quantum. </span><span face="sans-serif" style="background-color: white; font-size: 13px; line-height: 19.0469px;">For probabilistic systems we describe our configuration by a set of positive real numbers which sum to unity, while for quantum systems these numbers are replaced by amplitudes, complex numbers whose absolute value squared sum to unity. A quantum system is as simple as that: a system whose description is given by amplitudes and not probabilities.</span><br />
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<span face="sans-serif" style="background-color: white; font-size: 13px; line-height: 19.0469px;">Hence, in the complex plane, the possible amplitudes which sum to unity are in a superposition of 0 and 1 as represented by the following Argand diagram:</span><br />
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Physicists like to represent qubits using what they call "Dirac ket notation," in which the vector (α,β) becomes <img align="center" alt="" src="http://www.scottaaronson.com/cgi-bin/mimetex.cgi?\alpha%20|0\rangle%20+%20\beta%20|1\rangle" />. Here α is the <em>amplitude</em> of outcome |0〉, and β is the amplitude of outcome |1〉.<br />
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<span style="background-color: white;"><span face="sans-serif" style="font-size: x-small;"><span style="line-height: 19.0469px;">Therefore, with digital systems we should have 3 different types of bit information configuration, classical bits which are deterministic, probabilistic bits which are stochastic and quantum bits which are unitary.</span></span></span><br />
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Therefore the concept of a quantum bit is similar to a probabilistic bit in that the value is government by a output percentage but this is not a probability anymore, it is an amplitude of a probability for which of the coherent superposition of a quantum observable's eigenstates (alpha and beta) jumps to, which is given by a probabilistic law such that the probability of the system jumping to the state is proportional to the absolute value of the corresponding linear combination squared.<br />
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In a sense, we can consider probability theory as a subset of quantum theory as far as computation is concerned.<br />
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Many body systems of entangled states in quantum circuits are studied using <span style="background-color: white;">powerful numerical techniques, some of which were developed under the broad scope of quantum information theory without specifying a particular physical framework for building the circuit, i.e. superconducting, optical, quantum spin, ect.</span><br />
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<span style="background-color: white;">The Density Matrix Renormalization Group (DMRG) is one such technique which </span><span style="background-color: white;">applies the Numerical Renormalization Group (NRG) to quantum lattice many-body systems such as the Hubbard model of strongly correlated electrons, forming Cooper pairs in superconductors say, as well as being extended to a great variety of problems in all fields of physics and to quantum chemistry.</span><br />
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The DMRG algorithm has attracted significant attention as a robust quantum chemical approach to multireference electronic structure problems in which a large number of electrons have to be highly correlated in a large-size orbital space. It can be seen as a substitute for the exact diagonalization method that is able to diagonalize large-size Hamiltonian matrices. It comprises only a polynomial number of parameters and computational operations, but is able to involve a full set of the Slater determinants or electronic configurations in the Hilbert space, of which the size nominally scales exponentially with the number of active electrons and orbitals, allowing a compact representation of the wavefunction.<br />
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In applications to quantum lattice systems, such as in superconducting quantum circuits, DMRG consists in a systematic truncation of the system Hilbert space, keeping a small number of important states in a series of subsystems of increasing size to construct wave functions of the full system. In DMRG the states kept to construct a <b>renormalization group</b> transformation are the most probable eigenstates of a reduced <b>density matrix</b> instead of the lowest energy states kept in a standard NRG calculation. DMRG techniques for strongly correlated systems have been substantially improved and extended since their conception in 1992. They have proved to be both extremely accurate for low-dimensional problems and widely applicable. They enable numerically exact calculations (i.e., as good as exact diagonalizations) on large lattices with up to a few thousand particles and sites (compared to less than a few tens for exact diagonalizations).<br />
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Originally, DMRG has been considered as a renormalization group method. Recently, the interpretation of DMRG as a matrix-product state has been emphasized. From this point of view, DMRG is an algorithm for optimizing a variational wavefunction with the structure of a matrix-product state. This formulation of DMRG has revealed the deep connection between the density-matrix renormalization approach and quantum information theory and has lead to significant extensions of DMRG algorithms. In particular, efficient algorithms for simulating the time-evolution of quantum many-body systems have been developed.<br />
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Simulating large quantum circuits with a classical computer requires enormous computational power, because the number of quantum states increases exponentially as a function of number of qubits. The DMRG was introduced to study the properties of relatively large-scale one-dimensional quantum systems, as this method corresponds to an efficient data compression for one-dimensional quantum systems. By applying the DMRG method to quantum circuit simulation, simple quantum circuits based on Grover's algorithm can be simulated with a classical computer of reasonably modest computational power.<br />
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In such simulations, we are dealing with reversible quantum computing circuits, which requires a special type of bit, called an ancilla bit. In this scheme the ancilla bit must be prepared as a fixed qubit state used for input to a gate to give the gate a more specific logic function. These can be either prepared as computational basis of random numbers, possible solutions or guess heuristics.<br />
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DMRG works well on all physical systems with low ground state entanglement.<br />
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However QMA problems yield highly entangled ground states! To do DMRG models we would need less entangled ground states which would be an automatic restriction of proof & verifier circuit!<br />
Our only choice therefore would be to just accept classical proofs and classical verifier circuits, i.e. restriction of all problems to NP.<br />
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QMA: The quantum version of NP – the class of problems where “yes” instances have a quantum proof which can be efficiently checked by a quantum computer<br />
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We are interested in the spectral gap for each instance independently not in the promise gap between “yes” and “no” instances.<br />
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To make registering locally accessible time must be encoded in spatial location of qubits, by the time-dependent Hamiltonian.<br />
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Realization of Hamiltonian is made by adding a control register; implement a control unit, a <span style="color: red;">“head”</span>, of propagating qubits in the circuit.<br />
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A finite control unit consists of a finite number of qubits pi that the condition of the machine are in a Hilbert space.<br />
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An infinite memory of which a finite part is used: this is an infinite set of qubits mi serving in a Hilbert space and memory for the machine.<br />
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On the basis of the density matrix, rho, we can re-express the concept of observation as a probability.<br />
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The probability of finding a particular eigenstate, |k>, in the quantum register in a given initial state, |phi> can be calculated as<br />
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We can, in a similar way, get the expectation value of a quantum measurement, as<br />
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A is the observable with the matrix of the system.<br />
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This is nothing more than the average of individual probabilities of energy configuration values.<br />
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The energy expectation value will not be used further in this text, but it is important as an educational example for the NMR quantum computer which was the first use of this technology based on the quantum energy states of molecules under a particular spin state.<br />
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The NMR quantum computer, using a powerful external magnetic field to align molecules, made measurements using uniform microwave pulses and, at a measurement, then gets the average of all possible energy configurations, which correspond to whether the spins are aligned with or against the external field.<br />
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<span face="sans-serif" style="font-size: 13px; line-height: 19.05px;">Nuclear spins are advantageous as qubits, since they have very good isolation from mechanisms that can lead to decoherence. This leads to long relaxation times. Moreover spins can be manipulated by </span><img alt="RF\," class="tex" src="http://ictwiki.iitk.ernet.in/wiki/images/math/e/5/6/e56aaf777919287d1dd664075c18f15c.png" style="border: none; font-family: sans-serif; font-size: 13px; line-height: 19.05px; margin: 0px; vertical-align: middle;" /><span face="sans-serif" style="font-size: 13px; line-height: 19.05px;"> </span><span face="sans-serif" style="font-size: 13px; line-height: 19.05px;">pulses to effect appropriate unitary operations.</span><br />
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Tthe Hamiltonian of a spin <img alt="1/2\," class="tex" src="http://ictwiki.iitk.ernet.in/wiki/images/math/7/1/3/713a80fbe83412120afd2deff913f1b4.png" style="border: none; margin: 0px; vertical-align: middle;" /> particle in constant magnetic field <img alt="B_0\," class="tex" src="http://ictwiki.iitk.ernet.in/wiki/images/math/f/9/3/f932bce52e343a3a0d330566540ef4b2.png" style="border: none; margin: 0px; vertical-align: middle;" /> along <img alt="\hat z\," class="tex" src="http://ictwiki.iitk.ernet.in/wiki/images/math/7/8/4/784ffea1f74c6a5d5f9ecfab48e398ec.png" style="border: none; margin: 0px; vertical-align: middle;" />, is given by</div>
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<span face="sans-serif" style="font-size: 13px; line-height: 19.05px; text-align: left;">where</span><span face="sans-serif" style="font-size: 13px; line-height: 19.05px; text-align: left;"> </span><img alt="\omega_0\," class="tex" src="http://ictwiki.iitk.ernet.in/wiki/images/math/1/a/f/1af5f9d044c1dec5c58f879841621707.png" style="border: none; font-family: sans-serif; font-size: 13px; line-height: 19.05px; margin: 0px; text-align: left; vertical-align: middle;" /><span face="sans-serif" style="font-size: 13px; line-height: 19.05px; text-align: left;"> </span><span face="sans-serif" style="font-size: 13px; line-height: 19.05px; text-align: left;">is the Larmor precession frequency and is typically of the order of 100MHz. Spins of different nuclei are distinguishable by their different Larmor Frequencies. Spins of same nuclei in a molecule can have slightly different Larmor frequencies owing to different chemical shifts (typically in 10-100ppm), hence rendering them distinguishable</span></div>
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It is convenient to use the pictorial Bloch Sphere representation</div>
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<a href="http://ictwiki.iitk.ernet.in/wiki/images/math/6/1/a/61a0d73a0bdca7e7513758e6b6135aef.png" style="margin-left: 1em; margin-right: 1em;"><img alt="|\psi\big\rangle = cos(\theta/2)|0\big\rangle+sin (\theta/2)e^{i\phi}|1\big\rangle" border="0" class="tex" src="http://ictwiki.iitk.ernet.in/wiki/images/math/6/1/a/61a0d73a0bdca7e7513758e6b6135aef.png" style="border: none; vertical-align: middle;" /></a></div>
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<a class="image" href="http://ictwiki.iitk.ernet.in/wiki/index.php/File:Bloch.png" style="background-image: none; clear: left; color: #5a3696; float: left; margin-bottom: 1em; margin-right: 1em; text-decoration: none;" title="Bloch.png"><img alt="" border="0" height="259" src="http://ictwiki.iitk.ernet.in/wiki/images/thumb/Bloch.png/400px-Bloch.png" style="border: none; cursor: move; margin-left: auto; margin-right: auto; vertical-align: middle;" width="400" /></a><span face="sans-serif" style="font-size: 13px; line-height: 19.05px; text-align: left;">where</span><span face="sans-serif" style="font-size: 13px; line-height: 19.05px; text-align: left;"> </span><img alt="\theta\," class="tex" src="http://ictwiki.iitk.ernet.in/wiki/images/math/1/f/0/1f09c25c5247c1eaf121df644ca42f8c.png" style="border: none; font-family: sans-serif; font-size: 13px; line-height: 19.05px; margin: 0px; text-align: left; vertical-align: middle;" /><span face="sans-serif" style="font-size: 13px; line-height: 19.05px; text-align: left;"> </span><span face="sans-serif" style="font-size: 13px; line-height: 19.05px; text-align: left;">is the angle from the z-axis and</span><span face="sans-serif" style="font-size: 13px; line-height: 19.05px; text-align: left;"> </span><img alt="\phi\," class="tex" src="http://ictwiki.iitk.ernet.in/wiki/images/math/c/d/0/cd014731964c742c274df08d7cc238fb.png" style="border: none; font-family: sans-serif; font-size: 13px; line-height: 19.05px; margin: 0px; text-align: left; vertical-align: middle;" /><span face="sans-serif" style="font-size: 13px; line-height: 19.05px; text-align: left;"> </span><span face="sans-serif" style="font-size: 13px; line-height: 19.05px; text-align: left;">is the azimuthal angle. Under the constant magnetic field, the Bloch vector precesses about the z-axis.</span></div>
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Application of <img alt="RF\," class="tex" src="http://ictwiki.iitk.ernet.in/wiki/images/math/e/5/6/e56aaf777919287d1dd664075c18f15c.png" style="border: none; margin: 0px; vertical-align: middle;" /> field of amplitude <img alt="B_1 = \hbar\omega_1" class="tex" src="http://ictwiki.iitk.ernet.in/wiki/images/math/0/4/4/0441079e5ec7fdc1a294f253b9a841cf.png" style="border: none; margin: 0px; vertical-align: middle;" /> at <img alt="\omega_{rf} = \omega_0\," class="tex" src="http://ictwiki.iitk.ernet.in/wiki/images/math/9/2/5/92530f639f4a91199d113ccbec8b0c11.png" style="border: none; margin: 0px; vertical-align: middle;" />, makes the spin evolve under the transformation<br />
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<a href="https://blogger.googleusercontent.com/img/proxy/AVvXsEjdltUBRL47Z-_DQhx_ctibid8ca-J4d05jdIfQ04bSqC8fkH42mHfkPOl7ZiHnpBtaFLj_plytTh4D3qUgMBtpb9XSu45Yv1CoFaeU6ZKVy8-QxrPNbcWddc1J9wk1FU_pag3qHUlsJnYbZ3MGxwnlHZpgtsosozGCIltOlVdE0wZkhFyKsXJjzUY_wT-fbCcqYesxCfMZUA=" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img alt="U = e^{i\omega_1}(cos\phi I_x-sin \phi I_y)^t_{p\omega}" border="0" class="tex" src="http://ictwiki.iitk.ernet.in/wiki/images/math/6/2/7/627bc877841dd4fa7d45996cd56dc735.png" style="border: none; vertical-align: middle;" /></a><br />
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where <img alt="t_{p\omega}\," class="tex" src="http://ictwiki.iitk.ernet.in/wiki/images/math/a/b/1/ab19bd03827fdce9a2f5c25b18d80fb8.png" style="border: none; margin: 0px; vertical-align: middle;" /> is the pulse-width.</div>
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This gives a rotation by an angle proportional to the product of tpw and !1 about an axis in the <img alt="x y\," class="tex" src="http://ictwiki.iitk.ernet.in/wiki/images/math/4/0/5/405452f4658a2f6e5ffd68821b15e033.png" style="border: none; margin: 0px; vertical-align: middle;" /> plane determined by the phase <img alt="\phi\," class="tex" src="http://ictwiki.iitk.ernet.in/wiki/images/math/c/d/0/cd014731964c742c274df08d7cc238fb.png" style="border: none; margin: 0px; vertical-align: middle;" />. E.g. A pulse of <img alt="\omega_1 t_{p\omega}=\pi/2\," class="tex" src="http://ictwiki.iitk.ernet.in/wiki/images/math/c/b/4/cb4cb78c94712ea23e44940c46510c87.png" style="border: none; margin: 0px; vertical-align: middle;" /> and <img alt="\phi = \pi\," class="tex" src="http://ictwiki.iitk.ernet.in/wiki/images/math/2/2/a/22a4aff2f265e89d379e93e0a75dd37b.png" style="border: none; margin: 0px; vertical-align: middle;" /> cause rotation about <img alt="\hat x\," class="tex" src="http://ictwiki.iitk.ernet.in/wiki/images/math/3/6/5/365a712bb2f21c0c9e43efc0a905f986.png" style="border: none; margin: 0px; vertical-align: middle;" /> by <img alt="\pi/2\," class="tex" src="http://ictwiki.iitk.ernet.in/wiki/images/math/8/b/b/8bb2ddc7cd5a517d58cb2afc3c2e66a9.png" style="border: none; margin: 0px; vertical-align: middle;" />, i.e. <img alt="R_x\," class="tex" src="http://ictwiki.iitk.ernet.in/wiki/images/math/1/0/5/105babb217208c67dc8a0bcc76ecdef3.png" style="border: none; margin: 0px; vertical-align: middle;" /> (90), where as a pulse of same width but with phase causes <img alt="R_y\," class="tex" src="http://ictwiki.iitk.ernet.in/wiki/images/math/a/7/4/a74a0009bb3b9e0ca8b673e0e5fefbdb.png" style="border: none; margin: 0px; vertical-align: middle;" />.</div>
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Using rotations about <img alt="\hat x\," class="tex" src="http://ictwiki.iitk.ernet.in/wiki/images/math/3/6/5/365a712bb2f21c0c9e43efc0a905f986.png" style="border: none; margin: 0px; vertical-align: middle;" /> and <img alt="\hat y\," class="tex" src="http://ictwiki.iitk.ernet.in/wiki/images/math/6/2/8/6286b4a50d5ea958ae011480ef78f90f.png" style="border: none; margin: 0px; vertical-align: middle;" />, any arbitrary single qubit <img alt="U\," class="tex" src="http://ictwiki.iitk.ernet.in/wiki/images/math/6/f/3/6f3d5ad4b0e22c80e2db450cf7238ea9.png" style="border: none; margin: 0px; vertical-align: middle;" /> can be implemented</div>
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Using the spin-spin coupling Hamiltonian it is possible to impliment a 2-qubit C-NOT Gate.</div>
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Physically is can be said that, a spin \feels" an additional magnetic field due to neighboring spins causing a shift in Larmor frequency A line selective <img alt="\pi\," class="tex" src="http://ictwiki.iitk.ernet.in/wiki/images/math/e/a/d/eadbf26e3cb9eae366fbb8c8593ac9e9.png" style="border: none; margin: 0px; vertical-align: middle;" /> pulse at frequency, <img alt="\omega_0^2+\pi j" class="tex" src="http://ictwiki.iitk.ernet.in/wiki/images/math/5/9/5/595f846131191b62657804dd673dc724.png" style="border: none; margin: 0px; vertical-align: middle;" /> can then be used as a CNOT operation. This causes spin ip of the second qubit, via Rabi oscillation, only when the first qubit is 1.</div>
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Two qubit Control Not Gate</div>
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<span face="sans-serif" style="font-size: x-small;"><span style="line-height: 19.05px;">There are several hurdles in implementing NMR quantum computation. First of all it is extremely difficult to address and read individual spins. A large number of molecules must be present to produce measurable signals.The number of molecules in the 7-bit NMR quantum computer built by IBM for example had a magnitude of 10^18 molecules Hence the NMR signal is an average over all the molecules' signals. </span></span></div>
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<span face="sans-serif" style="font-size: 13px; line-height: 19.05px;">The quantum register used for the implementation is a organic molecule consisting of five </span><span class="texhtml" style="font-family: serif; font-size: 13px; line-height: 19.05px;"><sup>19</sup></span><span face="sans-serif" style="font-size: 13px; line-height: 19.05px;"> </span><span face="sans-serif" style="font-size: 13px; line-height: 19.05px;">F and two</span><span face="sans-serif" style="font-size: 13px; line-height: 19.05px;"> </span><span class="texhtml" style="font-family: serif; font-size: 13px; line-height: 19.05px;"><sup>13</sup></span><span face="sans-serif" style="font-size: 13px; line-height: 19.05px;"> </span><span face="sans-serif" style="font-size: 13px; line-height: 19.05px;">C nuclei, i.e. a total of seven spin-1/2 nuclei, as shown below:</span><br />
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<span face="sans-serif" style="font-size: x-small; line-height: 19.05px;">But quantum computation requires preparation, manipulation, coherent evolution and measurement of pure quantum states, so using a statistical mixture above a certain number of molecules eliminates the coherence required for quantum computation meaning that NMR quantum computation was limited using such approaches to move beyond a few 10s of qubits.</span><br />
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Regardless of the technology, with an observed quantum register we get, after detection, a new initial state given by<br />
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Note that we in our observation of a quantum register we project the state vector at a subspace of the Hilbert space:<br />
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Considering a 3-qubit quantum register<br />
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We first calculate the odds of calculating the values {0.....7} of the quantum register<br />
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After detection, the new initial state will then be<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjCwJ4qZloQDZ8xX5Uurxp_RuPwzO1HtyYvS9mqWP0B0yG1wxIYDjuAl1iFSSt5giGCSl3_nuwN8fEwaZidCWfPSzilceBV6L7du4ldKLc1w8Cr-iEK-h9nnYVYfhVB5jDSP1B12OnT_I3x/s1600/initial+state.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="170" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjCwJ4qZloQDZ8xX5Uurxp_RuPwzO1HtyYvS9mqWP0B0yG1wxIYDjuAl1iFSSt5giGCSl3_nuwN8fEwaZidCWfPSzilceBV6L7du4ldKLc1w8Cr-iEK-h9nnYVYfhVB5jDSP1B12OnT_I3x/s1600/initial+state.png" width="320" /></a></div>
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In the Matlab software, this quantum registering procedure can be implemented<br />
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function x = measure(x, k)<br />
dim = length(x); % the number of basis states<br />
n = log2(dim); % number of bits in the register<br />
if nargin == 1, k = [0:n-1]; end;<br />
for i = 1:length(k)<br />
prob_0 = probability_0(x, k(i));<br />
if prob_0 > rand<br />
% observed a ’0’<br />
x = apply([1/sqrt(prob_0) 0; 0 0], k(i), x);<br />
else<br />
% observed a ’1’<br />
x = apply([0 0; 0 1/sqrt(1-prob_0)], k(i), x);<br />
end;<br />
end;<br />
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dynamical correlation model, Canonical Transformation Theory, that is tailored to efficient incorporation into large-scale static (or strong) correlation. In CT model, higher-level dynamic correlations are described from a unitary cluster many-body operator,<br />
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eA |Ψ0 ⟩ = (1 + A + 1/2 A2+...) |Ψ0 ⟩, where A†=-A.<br />
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In a related picture, we can view eA as generating an effective canonically transformed Hamiltonian HCT that acts only in the active space, but which has dynamic correlation folded in from the external space, where<br />
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HCT = e-A H eA = H + [H,A] + 1/2 [[H,A],A] + ....<br />
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A central feature of the canonical transformation theory is the use of a novel operator decomposition, both to "approximately" close the infinite expansions associated with the exponential ansatz and to reduce the complexity of the energy and amplitude equations, resulting in a scalable internally-contracted multireference algorithm (o(N6)).<br />
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<img alt="ct" height="292" src="http://qcl.ims.ac.jp/img/research/ct1.png" width="400" /><br />
CT algorithm is designed to be incorporated into large-scale static correlation, which is assumed to be handled with modern powerful diagonalization techniques, such as density matrix renormalizatoin group (DMRG).<br />
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Expanding on these models generates the quantum analog of the Turing machine:<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEihG_wx5J5PKxyIpoYHPA8VZrHuJ12nZI3dXiKM9HJXqnM5V-C5yf3FysVnksm4yBoPm6ioRmCdKmlpaCR7D9NREf9FElDQJZBkoQQ7vEzOyNTWpzidFD0plG31yJr8rjvOL2bswi7U3pZx/s1600/qc_graph5.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="245" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEihG_wx5J5PKxyIpoYHPA8VZrHuJ12nZI3dXiKM9HJXqnM5V-C5yf3FysVnksm4yBoPm6ioRmCdKmlpaCR7D9NREf9FElDQJZBkoQQ7vEzOyNTWpzidFD0plG31yJr8rjvOL2bswi7U3pZx/s1600/qc_graph5.png" width="400" /></a></div>
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The resulting Hamiltonian has a spectral gap above ground states. Finding the ground state (energy) is NP-hard. Therefore other techniques must be developed to transform the initial ground state of a quantum system to the final ground state.<br />
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Adiabatic Quantum Computation (AQC) is a universal model for quantum computation which seeks to transform the initial ground state of a quantum system into a final ground state encoding the answer to a computational problem.<br />
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By using Josephson junctions, like those present in Superconducting Quantum Interferencd Devices (SQUIDs) as qubit elements in a quantum computer chip, such as constructed by NIST in America and D-Wave Systems in Canada, the algorithms used in Simulated Annealing can be extended into Quantum Annealing, where solutions exist that allow for the effects of quantum mechanics to tunnel through the maxima in probability distributions.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhADdoqMopR01VUhUdDf0jeyN2QqIiSVVYjefl02ZTp4-elcLlJ7v0UFbLgVbxebOGLDvipxvodutwE01f4S7NOoeYlY2CAGQapIoP7NtW3zSj-NSgpVXHwLkh5GiNUDlLtM7m1Hzkz9ila/s1600/squid.gif" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="205" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhADdoqMopR01VUhUdDf0jeyN2QqIiSVVYjefl02ZTp4-elcLlJ7v0UFbLgVbxebOGLDvipxvodutwE01f4S7NOoeYlY2CAGQapIoP7NtW3zSj-NSgpVXHwLkh5GiNUDlLtM7m1Hzkz9ila/s1600/squid.gif" width="320" /></a></div>
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You can have an analogy of the annealing by imagining you put a block of ice in a cup, and you turn up the heat to make it melt. Your goal is to make the ice melt as slowly as possible, so that when you have your cup of water, you have absolutely zero vibrations or waves. You can imagine that by some super-heating method that if the ice cube were to instantaneously turn into water, there would be waves going everywhere since the water would be rushing out to the walls of the cup. What you want to do is heat it slowly so that this never happens, not even in the slightest. Your tolerance is, say, 99%. If your ice cube is an Adiabatic Quantum Computer, AQC, vibrations in the final state (the water) are "excitations", which basically mean errors and, thus, giving you wrong answers. They may be close, but they wouldn't be optimal.<br />
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So you have this system of connected qubits in SQUID form is that you prepare this system with a magnetic field that is going a certain way. You then you slowly turn off your initial state while slowly turning on your final state. So basically you're going through a mixed state of initial and final energy, though by the end there will be basically no initial parts of the problem left in the state and you're left only with the final state. An alternative way is to start with both states on, but have the initial state be much, much stronger than the final part of the state, and then slowly turn off the very large, initial state. If you find this confusing, take a look at this equation<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEitklkmpaqpkrrbpUMIyzTga5pKAuwa-py2lyQ6DzPu3vsi3VgHmGzteML1JJ27T9c9E1KXmnZSBIyE3Lj1ASrPINp9fyUuy7qk0dBYyljpkidjiJ5L8Ji5uuJn5iuuncvRkp0FcujMjfDr/s1600/hamiltonian.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="37" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEitklkmpaqpkrrbpUMIyzTga5pKAuwa-py2lyQ6DzPu3vsi3VgHmGzteML1JJ27T9c9E1KXmnZSBIyE3Lj1ASrPINp9fyUuy7qk0dBYyljpkidjiJ5L8Ji5uuJn5iuuncvRkp0FcujMjfDr/s1600/hamiltonian.png" width="320" /></a></div>
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H stands for Hamiltonian, which is just a name for a matrix that defines the total energy state of the system. The three terms, you'll notice, can be grouped into two terms really. the big Z and X are referring to the spin matrices. So in terms of this equation, your initial state is governed by the last term, and your final state is governed by the first two terms. The spin matrices show the change from the X-basis state to the Z-basis state. When you prepare your initial system of qubits, you put them all into the X-spin state and then anneal to the Z-spin state. Imagine that h and J are much, much smaller than K initially. When you anneal, you slowly turn off K so that when you're done, you're only left with whatever is in the first two terms. This is how the annealing process works.<br />
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Now in order to anneal properly so that you get the correct answer, you need to be sure you're giving it enough time to settle without excitations. The above equation is time-dependent, or more specifically, the h, J, and K terms are time-dependent. Now the adiabatic theorem will tell you that unless you run for t = infinity, you will not reach 100% accuracy. Still, we can get close. Even 90% isn't so bad, given that you're quickly reaching a somewhat optimal solution.<br />
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In terms of the math and quantum mechanics, your state vector always needs to be in the lowest eigenstate of the Hamiltonian. An eigenstate, in linear algebra, would be equivalent to an eigenvector of some matrix. If you're familiar with quantum mechanics, you'll know that particles can only be in discrete energy states. The energy of this system would be the Hamiltonian, which is a matrix, and the lowest energy state is given by the lowest eigenvector of this Hamiltonian. If you wanted to know the actual energy level, that would be the lowest eigenvalue that goes with the aforementioned eigenvector/eigenstate. Now when you're annealing, you always want to be in the ground state. This is why you must move slowly, because if you move too quickly you are imparting energy into the system, causing excitations and jumps to higher energy levels. This is not good because that's where your errors come from.<br />
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Once you've gone through the annealing process, you'll wonder a few things. How do I know I've gotten the right answer? How much time should I have given my program to run? Is there a way to do some on-the-fly error correction?<br />
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Well, you will have to do tests on your problem to know if you got the right answer. Typically if you're solving an NP-complete problem, you can check if your solution is correct pretty quickly. Still, you will want to run the computation many times until you put that particular algorithm to use in any serious case. There are measures for this as well, however, and they have to do with tracking the actual energy of the Hamiltonian, finding its ground state eigenvector, and then comparing it with what your qubit state actually is. The inner product of the two will show you your error. Note that this can only be done theoretically, as you wouldn't be able to track your quantum state without observing it (and therefore, changing it).<br />
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On the fly error correction is also possible in a simulation of an AQC. Let me back up here. There is an idea that you can write some numerical software so that you can simulate the annealing process and come up with a time-scheme for your particular algorithm. Since it's a simulation, you have access to your state vector at all times and you can measure the predicted error according to the mathematics. You can do on-the-fly error correction by looking at the measures of error and correctness, and then adjusting your annealing timesteps accordingly. Take a look at this graph:<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgkker_n_35qQm_f4hFtkilvzfNiWT4bpn2_bp_-WkNnVNqoW7obrzwBDCjV6p5liyw90vg4VjlgwgqfoMDpm9AjrGQ6Ym7NTW-TIfLB_RmgumrHMBcIo0QCUVZXX9fGRr-3-F1HfGqdipy/s1600/eigenspectrum.gif" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgkker_n_35qQm_f4hFtkilvzfNiWT4bpn2_bp_-WkNnVNqoW7obrzwBDCjV6p5liyw90vg4VjlgwgqfoMDpm9AjrGQ6Ym7NTW-TIfLB_RmgumrHMBcIo0QCUVZXX9fGRr-3-F1HfGqdipy/s1600/eigenspectrum.gif" /></a></div>
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This is what is called the eigenspectrum of a 1-qubit simulation. Basically, it is showing you the energies through the annealing time for 1 qubit. Look at the x-axis as time. The bottom curve is the ground state, and the upper curve is the first excited state. Notice how at some point you will get that the two curves are very close. This is a crucial point, because the smaller this energy "gap" is, the more likely it is that the state of the system will jump to this higher energy state instead of stay in the ground state. Clearly this is a time where you want to move very slowly. Conversely, when the energy gap is very large, you can move rather quickly without worrying if you're going to jump to a higher state since it's pretty unlikely. You can come up with a time-scheme that means you go very fast up until that n = 0.5 point, and start to go much more slowly until you've passed the small-gap threshold.<br />
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By applying the concept of adiabatic transitions to maxima and minima in simulated annealing, it is possible to examine how particles could quantum tunnel between such maxima and minima.<br />
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Adiabatic curves exist between the maxima and minima of isotherms in thermodynamic plots. Such adiabatic transitions are, on the level of quantum mechanics, happen as competing a process where quantum systems tunnel between energy barriers.<br />
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<img height="312" src="http://www.organicdesign.co.nz/files/thumb/f/fd/Quantum_annealing.jpg/250px-Quantum_annealing.jpg" width="400" /><br />
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In superconductor materials, electrons form in pairs of particles, forming bosonic quasi-particles called Cooper pairs. These bosons can form entangled states with one another at temperatures where the thermal oscillations of the individual pairs are quenched, dropping the potential energy barrier to form such entanglement, therefore allowing the effects of quantum entanglement take over. This effectively forms a Bose-Einstein Condensate, where all particles are at the lowest possible energy level.<br />
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<img src="http://t1.gstatic.com/images?q=tbn:ANd9GcRNkAw9hMCMVib36ahjUnt-xgLcycGRs775FumPhfShHWBkpvLaSQ" /><br />
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By incorporating superconducting materials in Josephson Junction circuits such that quanta can tunnel between the energy barriers experienced at the boundary between the superconductor and insulator or normal metal in an adiabatic mode, quantum switching elements can be created, and hence quantum computers, can be built around such models.<br />
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<img height="256" src="http://www.pnas.org/content/107/28/12446/F1.large.jpg" width="400" /><br />
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Schematic representation of a level anticrossing. The energies of two quantum states |Ψ1〉 and |Ψ2〉 localized in distant wells can be fine-tuned by applying a smooth additional potential.<br />
(A) Before the crossing, the ground state is |Ψ2〉 with energy close to E2(s); i.e., for s- < sc, we have that E1(s-) > E2(s-), so that |GS(s-)〉 = |Ψ2〉.<br />
(B) After the crossing, the ground state becomes |Ψ1〉 with energy close to E1(s); i.e., for s+ > sc, we have that E1(s+) < E2(s+), so that |GS(s+)〉 = |Ψ1〉.<br />
The ground states before and after the crossing have nothing to do with each other. At a certain interval of s close to sc, the anticrossing takes place and the ground state is a linear combination of |Ψ1〉 and |Ψ2〉.<br />
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The nature of these models is based on the Hamiltonian operator derived from the adiabatic theorem.<br />
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The main idea behind quantum adiabatic algorithms is to employ the Schrödinger equation<br />
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AQC algorithms involve the specification of a time- dependent Hamiltonian,<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi8LOuxNCZ2soCG25wNxCrkm27c6S4NXnamA70J8BqGOlki2M6CvxvQ-fcJrF4ZD6pSlZYZb59RIeqFoqdLlmVkRYooiNZ12PeBRAzDj5bQze4QiyBQEKpcZMtfbzxqGg1aDUTWUhH_wC-z/s1600/Hamiltonian.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="236" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi8LOuxNCZ2soCG25wNxCrkm27c6S4NXnamA70J8BqGOlki2M6CvxvQ-fcJrF4ZD6pSlZYZb59RIeqFoqdLlmVkRYooiNZ12PeBRAzDj5bQze4QiyBQEKpcZMtfbzxqGg1aDUTWUhH_wC-z/s1600/Hamiltonian.png" width="400" /></a></div>
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This Hamiltonian has three important functions: </div>
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(1) The initial Hamiltonian, Hi ≡ H(0), encodes a ground state that is easy to prepare and that is used as the initial state for the quantum evolution. </div>
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(2) The driving Hamiltonian, ˆ hdriving(t), is responsible for mediating the transformation of the initial ground state to any of other state. </div>
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(3) The final Hamiltonian, ˆ Hf ≡ ˆ H(τ), is prob- lem dependent and its ground state encodes the solu- tion, |ψsolutioni, to the computational problem. </div>
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In the ideal case of a process being fully adiabatic, evolution under ˆ H(t) will keep the quantum state, |ψ(t)i, in the ground state of ˆ H(t) throughout 0 < t < τ. If this condition is met, the final state at t = τ should coin- cide with the ground state of the final Hamiltonian, ˆ Hf, i.e., |ψ(τ)i = |ψsolutioni, if the process is adiabatic. The measurement at t = τ will provide the solution to the computational problem.</div>
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Sombrero Adiabatic Quantum Computation (SAQC) Hamiltonian.<br />
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The goal of AQC algorithms is that of transforming an initial ground state |ψ(0)i into a final ground state |ψ(τ)i, which encodes the answer to the problem. This is achieved by evolving the corresponding physical sys- tem according to the Schr¨odinger equation with a time- dependent Hamiltonian ˆ H(t). </div>
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The AQC algorithm re- lies on the quantum adiabatic theorem [25–35], which states that if the quantum evolution is initialized with the ground state of the initial Hamiltonian, the time prop- agation of this quantum state will remain very close to the instantaneous ground state |ψg(t)i for all t ∈ [0,τ], whenever ˆ H(t) varies slowly throughout the propagation time t ∈ [0,τ]. This holds under the assumption that the ground state manifold does not cross the energy lev- els which lead to excited states of the final Hamiltonian. Here, we denote by ground state manifold the first m curves associated with the lowest eigenvalue of the time- dependent Hamiltonian for t ∈ [0,τ], where m is the degeneracy of the final Hamiltonian ground state. </div>
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Conventionally the adiabatic evolution path is the lin- ear sweep of s ∈ [0,1], where s = t/τ:</div>
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H(s) = (1−s)Htransverse + sHf. (3) ˆ</div>
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Htransverse (see Eq. 9 below) is usually chosen such that its ground state is a uniform superposition of all possible 2n computational basis vectors, for the case of an n−qubit system. Here, we choose the spin states {|qi = 0i,|q = 1i}, which are the eigenvec- tors of ˆ σz i with eigenvalues +1 and -1, respectively, as the basis vectors. </div>
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AQC initial Hamiltonians conventionally have a uniform superposition as ground state. A divergence from this practice can be made by introducing a simple form of heuristics: the ability to start the quantum evolution with a state which is a guess to the solution of the problem.</div>
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As preparing an arbitrary initial non-degenerate ground state for adiabatic evolution is not a trivial task, we focus on easy to prepare initial guesses that consist of one of the states in the computational basis. The strat- egy proposed builds initial Hamiltonians such that the initial guess corresponds to the non-degenerate ground state of the initial Hamiltonian, as it is required by AQC. Additionally, this ground state would be non-degenerate. Let us denote the states of the computational basis of an N qubit system as |qNi|qN−1i···|q1i ≡ |qN ···q1i where qn ∈ {0,1}. The proposed initial Hamiltonian, whose ground state corresponds to an arbitrary initial guess state of the form |xN ···x1i, can be written as ˆ</div>
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Hi =</div>
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N X n=1</div>
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xn ˆ I + ˆ qn(1−2xn) =</div>
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N X n=1 ˆ</div>
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hxn, (5)</div>
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where each xn is a boolean variable, xn ∈{0,1}, while ˆ q ≡ 1 2(ˆ I − ˆ σz) is a quantum operator acting on the n-th qubit of the multipartite Hilbert space HN ⊗HN−1 ⊗ ···⊗Hn ⊗···⊗H1. The operator ˆ qn is given by ˆ</div>
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qn = ˆ IN ⊗ ˆ IN−1 ⊗···⊗(ˆ q)n ⊗···⊗ ˆ I1, (6)</div>
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where ˆ q is placed in the nth position and the identity operators act on the rest of the Hilbert space. The states constituting the computational basis, |0iand |1i, are eigenvectors of ˆ σz with eigenvalues +1 and −1, and therefore they are also eigenstates of the oper- ator ˆ q with eigenvalues 0 and 1 respectively. The logic behind the initial Hamiltonian in Eq. 5 then is clear: if xn = 0, then ˆ hxn=0 = ˆ qn but in the case of xn = 1, then ˆ hxn=1 = ˆ I − ˆ qn.</div>
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a sig- nificant amount of progress has been made towards the design of final Hamiltonians for different computation- ally intractable problems such as NP-complete problems. </div>
construction of the final Hamiltonian for an NP- hard problem of interest in biology, such as in the protein folding problem, a result of the famous Thermodynamic Hypothesis of protein Folding, which consists of finding the minimum energy configuration of a chain of interacting amino acids in a lattice model.<br />
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The hypotheisis is that the native fold of a globular protein is usually assumed to correspond to the global minimum of the protein’s Gibbs free energy.<br />
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The protein folding problem can be thus analyzed as a global optimization problem, with global maxima and minima.<br />
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<img height="328" src="http://scop.berkeley.edu/thumbs/1.67/iz/d1izqa_/dl.png" width="400" /><br />
<span style="background-color: #f9f9f9;"><span face="sans-serif"><span style="font-size: 12px; line-height: 19.2px;">3D ribbon model of the protein ribonuclease A. </span></span></span><span face="sans-serif" style="background-color: #f9f9f9; font-size: 12px; line-height: 19.2px;">Beta strands are arrows and alpha-helices are spirals. </span><span face="sans-serif"><span style="font-size: 12px; line-height: 19.2px;">The disulphide bonds can from only after the protein folds into its native conformation. </span></span><span face="sans-serif" style="background-color: #f9f9f9; font-size: 12px; line-height: 19.2px;">The native folding of the protein could be modeled</span><span face="sans-serif" style="background-color: #f9f9f9; font-size: 12px; line-height: 19.2px;"> using global optimization models using Adiabatic Quantum Computation.</span><br />
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The main idea behind Sombrero Adiabatic Quantum Computation (SAQC) is to introduce heuristics in AQC, and having the possibility of restarting a failed AQC run from the measured excited state. In order to prepare an arbitrary state from any of the 2N possible basis states from the computational basis of the N qubit system, we propose an initial Hamiltonian, ˆ Hi (see Eq. 5), in such a way that the desired initial guess state is the non-degenerate ground state of the designed initial Hamiltonian. The initial Hamiltonian is diagonal in the computational basis, and so is the final Hamiltonian for the case of classical problems such as the NP-complete problems, e.g., random 3-SAT. Since both, the initial and final Hamiltonians are diagonal, connecting them via a linear ramp as is usually done in CAQC (see left panel) will not lead the quantum evolution towards finding the ground state of the final Hamiltonian. To maintain the initial Hamiltonian uniquely and fully turned on at the beginning, t = 0, and the final Hamiltonian uniquely and fully turned on at the end of the computation, t = τ, we introduce a driving Hamiltonian whose time profile intensity has a “sombrero-like” shape (see right panel) is such a way that it only acts during 0 < t < τ. Two examples of functions with this functional form are presented, where hat1(s) = sin2(πs) and hat2(s) = s(1−s). A desired feature of our algorithmic strategy is the possibility of introducing heuristics, and not that of introducing non-linear paths. The latter has been proposed previous publications [8, 36, 37], but here is employed as a consequence of the algorithmic strategy.<br />
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For SAQC, the time-dependent Hamiltonian can be written as: ˆ Hsombrero = (1−s) ˆ Hi + hat(s) ˆ Hdriving + s ˆ Hf. (4)<br />
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We want the non-degenerate ground state of the ini- tial Hamiltonian ˆ Hi to encode a guess to the solution, and the driving term, ˆ Hdriving, to couple the states in the computational basis. The function hat(s) is zero at the beginning and end of the adiabatic path; therefore ˆ Hdriving acts only in the range s ∈ (0,1) in a “sombrero mexican hat- like” time dependence<br />
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Implementation of an SAQC algorithm either in <span style="color: red;">parallel (for 2 or more quantum computers) </span>or in <span style="color: blue;">serial (for single quantum computers).</span><br />
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The algorithm begins by choosing a state from the computational basis. For each chosen initial state, an initial Hamiltonian is prepared based on a random or classical heuristic in a continuous cycle of operations.<br />
Next, an ideal time constraint is chosen, assuming a CAQC protocol will be run, which is to be used as a reference to run the SAQC protocol twice as fast.<br />
If only one AQC computer is available, we can cheat here by having a probabilistic model of SAQC to run two separate adiabatic protocols instead of one, in serial mode. Once the first SAQC calculation is finished, one can efficiently check whether or not the result is a solution. In case that it is not a solution, one can submit an additional calculation, either randomly selecting another initial guess state or using the measured excited state.<br />
If we use the measured excited state we are using a <span style="color: orange;">“quantum heuristic”</span> since the outcome of the near-adiabatic quantum evolution is used to refine the initial guess state for further use of the protocol.<br />
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In the case of having several adiabatic quantum computers at hand, one can do the same initial procedure of selecting guesses, but now submitting a different guess to a different node and running on each node twice as fast.<br />
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With this goal in mind, the viability of this approach has been explored along with the needed modifications to the conventional AQC (CAQC) algorithm. By performing a numerical study on hard-to-satisfy 6 and 7 bit random instances of the satisfiability problem (3-SAT), heuristic approaches are possible.<br />
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Scheme for 6 binary variables SAQC calculations. We generated 26 3-SAT unique satisfying assignment (USA) instances (first branching), each having as its only solution one of the 26 possible assignments. All 26 instances have a different state as solution, i.e. there is no chance for repeated instances. For each instance, we computed minimum-gap values associated with all possible settings of SAQC.<br />
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Of all possible guesses (second branching), using 20 different values of δ ∈{0.5,1.0,...,10.0} (third branching).<br />
The same scheme was applied to 7 binary variable 3-SAT USA instances (not shown) for a total of (128USA)×(128guesses)×(20values of δ) = 327,680 SAQC settings<br />
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The performance of the particular algorithm proposed is largely determined by the Hamming distance of the chosen initial guess state with respect to the solution. Besides the possibility of introducing educated guesses as initial states, the new strategy allows for the possibility of restarting a failed adiabatic process from the measured excited state as opposed to restarting from the full superposition of states as in CAQC. The outcome of the measurement can be used as a more refined guess state to restart the adiabatic evolution. This concatenated restart process is another heuristic that the CAQC strategy cannot capture.<br />
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By using a decentralized, quantum swarm AI, it may be possible to simulate emergent behavior aswell, such as Langton's ant, which could see the rise of quantum-based simulated intelligence which could go so far as to create realistic cellular automatons on a computer.<br />
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The use of sophisticated metaheuristics on lower heuristical functions may see computer simulations which can select specific sub routines on computers by themselves to solve problems in an intelligent way. In this way the machines would be far more adaptable to changes in sensory data and would be able to function with far more automation than would be possible with normal computers.<br />
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Moreover, it may be possible to use metaheuristics to perform error correction on software using neural networks by comparing the optimized problem solving in a quantum computer with the regular programming software of a regular computer. Since regular computers are not quantum mechanical, they must be programmed classically. However, by using quantum metaheuristics it may be possible to perform optimization problems using Artificial intelligence on a quantum computer and then compare to the command line architecture in a piece of conventional software on a classical computer, which may be too complex to modify or to check for errors using human software engineers.<br />
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It may become routine that to design large software, rather than use a team of engineers to design a single piece of large software and to do multiple tests, revisions and redistribution, quantum optimization problems will be run though a machine instructed to do a particular task and the software will be designed around such optimizations.<br />
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Computer chips could even be designed to be superconducting and quantum compatible in a factory or laboratory and classically operating in conventional computer equipment. In this way, optimization and programming could be done on the chips in a factory or lab to search for any possible errors that could arise in designing software. After which the chips would be used in conventional computers for their particular tasks. Quantum computers may well find their place in industry more than anywhere else, designing AI optimized software around computer chips.<br />
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Metaheuristics will depend on the existence of the software to be designed around a particular problem solution, hence its optimization of solving the solution comes from a search of such solutions in a network. such searches could utilized neural network algorithms and would be the simulation of artificial intelligence. However for true artificial intelligence another step must be made, one that goes back to the fundamental concept of a heuristic being a solution to optimization. The program must recognize that some heuristics under certain circumstances are good, while others are bad.<br />
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Such an intelligence would require automated planning, and would be essentially the computer controlling its own strategies, for use at certain times, based on predictive modelling. It would use these plans in automated strategies which would attempt to find the right method by selecting a sequence of learned metaheuristics in a given situation.<br />
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This would be no longer metaheuristics, it would be hyperheuristics. the fact that this species of heuristics depends on learned metaheuristics means that it depends on feed-back learning. Hence this would be a machine that, after an initial amount of metaheuristic training would eventially begin to learn itself in a sulf-sustained way. In this way, a hyperheuristical machine would start out in a metaheuristical state and after a period of training it would "evolve" into a hyperheuriscal machine and begin a learning feedback.<br />
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How could it accomplish this? Hyperheursics could be to have the computer make changes to the theoretical global optimization model in response to learned behavior which would crate a sustained loop between learning and decision making so that it would have complete automated planning in response to an input. An adaptive planning machine would have to be a hyperheuristical machine.<br />
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As we have seen, the limitations of global optimization models can be improved through the technique of Adiabatic Quantum Computation. Therefore the most advanced type of heuristic possible would have to be based on quantum computer technology.</div>
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Approximations of such AI come in the form of Memetic Algorithms which evolve under the concept of Universal Darwinism. These programs are evolved on the basis of the transference of information.</div>
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<span face="sans-serif" style="background-color: white; font-size: 12.7273px; line-height: 19.1903px;">applications include (but are not limited to) training of </span><a href="http://en.wikipedia.org/wiki/Artificial_neural_network" style="background-color: white; background-image: none; color: #0b0080; font-family: sans-serif; font-size: 12.7273px; line-height: 19.1903px; text-decoration: none;" title="Artificial neural network">artificial neural networks</a><span face="sans-serif" style="background-color: white; font-size: 12.7273px; line-height: 19.1903px;">,</span><sup class="reference" id="cite_ref-training_ANN_17-0" style="background-color: white; font-family: sans-serif; line-height: 1em; unicode-bidi: -webkit-isolate;"><a href="http://en.wikipedia.org/wiki/Memetic_algorithm#cite_note-training_ANN-17" style="background-image: none; color: #0b0080; text-decoration: none; white-space: nowrap;">[17]</a></sup><span face="sans-serif" style="background-color: white; font-size: 12.7273px; line-height: 19.1903px;"> </span><a href="http://en.wikipedia.org/wiki/Pattern_recognition" style="background-color: white; background-image: none; color: #0b0080; font-family: sans-serif; font-size: 12.7273px; line-height: 19.1903px; text-decoration: none;" title="Pattern recognition">pattern recognition</a><span face="sans-serif" style="background-color: white; font-size: 12.7273px; line-height: 19.1903px;">,</span><sup class="reference" id="cite_ref-pattern_recognition_18-0" style="background-color: white; font-family: sans-serif; line-height: 1em; unicode-bidi: -webkit-isolate;"><a href="http://en.wikipedia.org/wiki/Memetic_algorithm#cite_note-pattern_recognition-18" style="background-image: none; color: #0b0080; text-decoration: none; white-space: nowrap;">[18]</a></sup><span face="sans-serif" style="background-color: white; font-size: 12.7273px; line-height: 19.1903px;"> robotic </span><a href="http://en.wikipedia.org/wiki/Motion_planning" style="background-color: white; background-image: none; color: #0b0080; font-family: sans-serif; font-size: 12.7273px; line-height: 19.1903px; text-decoration: none;" title="Motion planning">motion planning</a><span face="sans-serif" style="background-color: white; font-size: 12.7273px; line-height: 19.1903px;">,</span><sup class="reference" id="cite_ref-motion_planning_19-0" style="background-color: white; font-family: sans-serif; line-height: 1em; unicode-bidi: -webkit-isolate;"><a href="http://en.wikipedia.org/wiki/Memetic_algorithm#cite_note-motion_planning-19" style="background-image: none; color: #0b0080; text-decoration: none; white-space: nowrap;">[19]</a></sup><span face="sans-serif" style="background-color: white; font-size: 12.7273px; line-height: 19.1903px;"> </span><a href="http://en.wikipedia.org/wiki/Charged_particle_beam" style="background-color: white; background-image: none; color: #0b0080; font-family: sans-serif; font-size: 12.7273px; line-height: 19.1903px; text-decoration: none;" title="Charged particle beam">beam</a><span face="sans-serif" style="background-color: white; font-size: 12.7273px; line-height: 19.1903px;"> orientation,</span><sup class="reference" id="cite_ref-beam_orientation_20-0" style="background-color: white; font-family: sans-serif; line-height: 1em; unicode-bidi: -webkit-isolate;"><a href="http://en.wikipedia.org/wiki/Memetic_algorithm#cite_note-beam_orientation-20" style="background-image: none; color: #0b0080; text-decoration: none; white-space: nowrap;">[20]</a></sup><span face="sans-serif" style="background-color: white; font-size: 12.7273px; line-height: 19.1903px;"> </span><a href="http://en.wikipedia.org/wiki/Circuit_design" style="background-color: white; background-image: none; color: #0b0080; font-family: sans-serif; font-size: 12.7273px; line-height: 19.1903px; text-decoration: none;" title="Circuit design">circuit design</a><span face="sans-serif" style="background-color: white; font-size: 12.7273px; line-height: 19.1903px;">,</span><sup class="reference" id="cite_ref-circuit_design_21-0" style="background-color: white; font-family: sans-serif; line-height: 1em; unicode-bidi: -webkit-isolate;"><a href="http://en.wikipedia.org/wiki/Memetic_algorithm#cite_note-circuit_design-21" style="background-image: none; color: #0b0080; text-decoration: none; white-space: nowrap;">[21]</a></sup><span face="sans-serif" style="background-color: white; font-size: 12.7273px; line-height: 19.1903px;"> electric service restoration,</span><sup class="reference" id="cite_ref-service_restoration_22-0" style="background-color: white; font-family: sans-serif; line-height: 1em; unicode-bidi: -webkit-isolate;"><a href="http://en.wikipedia.org/wiki/Memetic_algorithm#cite_note-service_restoration-22" style="background-image: none; color: #0b0080; text-decoration: none; white-space: nowrap;">[22]</a></sup><span face="sans-serif" style="background-color: white; font-size: 12.7273px; line-height: 19.1903px;"> medical </span><a href="http://en.wikipedia.org/wiki/Expert_system" style="background-color: white; background-image: none; color: #0b0080; font-family: sans-serif; font-size: 12.7273px; line-height: 19.1903px; text-decoration: none;" title="Expert system">expert systems</a><span face="sans-serif" style="background-color: white; font-size: 12.7273px; line-height: 19.1903px;">,</span><sup class="reference" id="cite_ref-medical_expert_system_23-0" style="background-color: white; font-family: sans-serif; line-height: 1em; unicode-bidi: -webkit-isolate;"><a href="http://en.wikipedia.org/wiki/Memetic_algorithm#cite_note-medical_expert_system-23" style="background-image: none; color: #0b0080; text-decoration: none; white-space: nowrap;">[23]</a></sup><span face="sans-serif" style="background-color: white; font-size: 12.7273px; line-height: 19.1903px;"> </span><a class="mw-redirect" href="http://en.wikipedia.org/wiki/Single_machine_scheduling" style="background-color: white; background-image: none; color: #0b0080; font-family: sans-serif; font-size: 12.7273px; line-height: 19.1903px; text-decoration: none;" title="Single machine scheduling">single machine scheduling</a><span face="sans-serif" style="background-color: white; font-size: 12.7273px; line-height: 19.1903px;">,</span><sup class="reference" id="cite_ref-single_machine_sched_24-0" style="background-color: white; font-family: sans-serif; line-height: 1em; unicode-bidi: -webkit-isolate;"><a href="http://en.wikipedia.org/wiki/Memetic_algorithm#cite_note-single_machine_sched-24" style="background-image: none; color: #0b0080; text-decoration: none; white-space: nowrap;">[24]</a></sup><span face="sans-serif" style="background-color: white; font-size: 12.7273px; line-height: 19.1903px;"> automatic timetabling (notably, the timetable for the </span><a class="mw-redirect" href="http://en.wikipedia.org/wiki/NHL" style="background-color: white; background-image: none; color: #0b0080; font-family: sans-serif; font-size: 12.7273px; line-height: 19.1903px; text-decoration: none;" title="NHL">NHL</a><span face="sans-serif" style="background-color: white; font-size: 12.7273px; line-height: 19.1903px;">),</span><sup class="reference" id="cite_ref-nhl_timetabling_25-0" style="background-color: white; font-family: sans-serif; line-height: 1em; unicode-bidi: -webkit-isolate;"><a href="http://en.wikipedia.org/wiki/Memetic_algorithm#cite_note-nhl_timetabling-25" style="background-image: none; color: #0b0080; text-decoration: none; white-space: nowrap;">[25]</a></sup><span face="sans-serif" style="background-color: white; font-size: 12.7273px; line-height: 19.1903px;"> </span><a href="http://en.wikipedia.org/wiki/Schedule_(workplace)" style="background-color: white; background-image: none; color: #0b0080; font-family: sans-serif; font-size: 12.7273px; line-height: 19.1903px; text-decoration: none;" title="Schedule (workplace)">manpower scheduling</a><span face="sans-serif" style="background-color: white; font-size: 12.7273px; line-height: 19.1903px;">,</span><sup class="reference" id="cite_ref-nurse_rostering_26-0" style="background-color: white; font-family: sans-serif; line-height: 1em; unicode-bidi: -webkit-isolate;"><a href="http://en.wikipedia.org/wiki/Memetic_algorithm#cite_note-nurse_rostering-26" style="background-image: none; color: #0b0080; text-decoration: none; white-space: nowrap;">[26]</a></sup><span face="sans-serif" style="background-color: white; font-size: 12.7273px; line-height: 19.1903px;"> </span><a class="new" href="http://en.wikipedia.org/w/index.php?title=Nurse_rostering_and_function_optimisation&action=edit&redlink=1" style="background-color: white; background-image: none; color: #a55858; font-family: sans-serif; font-size: 12.7273px; line-height: 19.1903px; text-decoration: none;" title="Nurse rostering and function optimisation (page does not exist)">nurse rostering and function optimisation</a><span face="sans-serif" style="background-color: white; font-size: 12.7273px; line-height: 19.1903px;">,</span><sup class="reference" id="cite_ref-nurse_rostering_function_opt_27-0" style="background-color: white; font-family: sans-serif; line-height: 1em; unicode-bidi: -webkit-isolate;"><a href="http://en.wikipedia.org/wiki/Memetic_algorithm#cite_note-nurse_rostering_function_opt-27" style="background-image: none; color: #0b0080; text-decoration: none; white-space: nowrap;">[27]</a></sup><span face="sans-serif" style="background-color: white; font-size: 12.7273px; line-height: 19.1903px;"> </span><a class="new" href="http://en.wikipedia.org/w/index.php?title=Processor_allocation&action=edit&redlink=1" style="background-color: white; background-image: none; color: #a55858; font-family: sans-serif; font-size: 12.7273px; line-height: 19.1903px; text-decoration: none;" title="Processor allocation (page does not exist)">processor allocation</a><span face="sans-serif" style="background-color: white; font-size: 12.7273px; line-height: 19.1903px;">,</span><sup class="reference" id="cite_ref-proc_alloc_28-0" style="background-color: white; font-family: sans-serif; line-height: 1em; unicode-bidi: -webkit-isolate;"><a href="http://en.wikipedia.org/wiki/Memetic_algorithm#cite_note-proc_alloc-28" style="background-image: none; color: #0b0080; text-decoration: none; white-space: nowrap;">[28]</a></sup><span face="sans-serif" style="background-color: white; font-size: 12.7273px; line-height: 19.1903px;"> maintenance scheduling (for example, of an electric distribution network),</span><sup class="reference" id="cite_ref-planned_maintenance_29-0" style="background-color: white; font-family: sans-serif; line-height: 1em; unicode-bidi: -webkit-isolate;"><a href="http://en.wikipedia.org/wiki/Memetic_algorithm#cite_note-planned_maintenance-29" style="background-image: none; color: #0b0080; text-decoration: none; white-space: nowrap;">[29]</a></sup><span face="sans-serif" style="background-color: white; font-size: 12.7273px; line-height: 19.1903px;">multidimensional knapsack problem,</span><sup class="reference" id="cite_ref-mkp_ma_30-0" style="background-color: white; font-family: sans-serif; line-height: 1em; unicode-bidi: -webkit-isolate;"><a href="http://en.wikipedia.org/wiki/Memetic_algorithm#cite_note-mkp_ma-30" style="background-image: none; color: #0b0080; text-decoration: none; white-space: nowrap;">[30]</a></sup><span face="sans-serif" style="background-color: white; font-size: 12.7273px; line-height: 19.1903px;"> </span><a class="mw-redirect" href="http://en.wikipedia.org/wiki/VLSI" style="background-color: white; background-image: none; color: #0b0080; font-family: sans-serif; font-size: 12.7273px; line-height: 19.1903px; text-decoration: none;" title="VLSI">VLSI</a><span face="sans-serif" style="background-color: white; font-size: 12.7273px; line-height: 19.1903px;"> design,</span><sup class="reference" id="cite_ref-vlsi_design_31-0" style="background-color: white; font-family: sans-serif; line-height: 1em; unicode-bidi: -webkit-isolate;"><a href="http://en.wikipedia.org/wiki/Memetic_algorithm#cite_note-vlsi_design-31" style="background-image: none; color: #0b0080; text-decoration: none; white-space: nowrap;">[31]</a></sup><span face="sans-serif" style="background-color: white; font-size: 12.7273px; line-height: 19.1903px;"> </span><a href="http://en.wikipedia.org/wiki/Cluster_analysis" style="background-color: white; background-image: none; color: #0b0080; font-family: sans-serif; font-size: 12.7273px; line-height: 19.1903px; text-decoration: none;" title="Cluster analysis">clustering</a><span face="sans-serif" style="background-color: white; font-size: 12.7273px; line-height: 19.1903px;"> of </span><a class="mw-redirect" href="http://en.wikipedia.org/wiki/Expression_profiling" style="background-color: white; background-image: none; color: #0b0080; font-family: sans-serif; font-size: 12.7273px; line-height: 19.1903px; text-decoration: none;" title="Expression profiling">gene expression profiles</a><span face="sans-serif" style="background-color: white; font-size: 12.7273px; line-height: 19.1903px;">,</span><sup class="reference" id="cite_ref-clustering_gene_expression_32-0" style="background-color: white; font-family: sans-serif; line-height: 1em; unicode-bidi: -webkit-isolate;"><a href="http://en.wikipedia.org/wiki/Memetic_algorithm#cite_note-clustering_gene_expression-32" style="background-image: none; color: #0b0080; text-decoration: none; white-space: nowrap;">[32]</a></sup><span face="sans-serif" style="background-color: white; font-size: 12.7273px; line-height: 19.1903px;"> feature/gene selection,</span><sup class="reference" id="cite_ref-gene_selection1_33-0" style="background-color: white; font-family: sans-serif; line-height: 1em; unicode-bidi: -webkit-isolate;"><a href="http://en.wikipedia.org/wiki/Memetic_algorithm#cite_note-gene_selection1-33" style="background-image: none; color: #0b0080; text-decoration: none; white-space: nowrap;">[33]</a></sup><sup class="reference" id="cite_ref-gene_selection2_34-0" style="background-color: white; font-family: sans-serif; line-height: 1em; unicode-bidi: -webkit-isolate;"><a href="http://en.wikipedia.org/wiki/Memetic_algorithm#cite_note-gene_selection2-34" style="background-image: none; color: #0b0080; text-decoration: none; white-space: nowrap;">[34]</a></sup><span face="sans-serif" style="background-color: white; font-size: 12.7273px; line-height: 19.1903px;"> and multi-class, multi-objective </span><a href="http://en.wikipedia.org/wiki/Feature_selection" style="background-color: white; background-image: none; color: #0b0080; font-family: sans-serif; font-size: 12.7273px; line-height: 19.1903px; text-decoration: none;" title="Feature selection">feature selection</a><span face="sans-serif" style="background-color: white; font-size: 12.7273px; line-height: 19.1903px;">.</span><sup class="reference" id="cite_ref-feature_selection_35-0" style="background-color: white; font-family: sans-serif; line-height: 1em; unicode-bidi: -webkit-isolate;"><a href="http://en.wikipedia.org/wiki/Memetic_algorithm#cite_note-feature_selection-35" style="background-image: none; color: #0b0080; text-decoration: none; white-space: nowrap;">[35]</a></sup></div>
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<h2>
Adaptive Heuristics - A Road to Machine Artificial Intelligence?</h2>
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Alan Turing was not only the founder of modern computer technology, based on switching elements be they valves, transistors, molecular states or quantum spin states, but he also wrote the first paper on Artificial Intelligence, AI.<br />
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Turing also thought we may instead change our concept of thinking, however in reality the study of artificial intelligence has not formulated any fundamental theories of the nature of thought processes. in the 50 years since Turing, AI procedures, with very few exceptions, have gone more towards accumulating known methods and strategies, i.e. heuristics and running through them rapidly with group force methods rather than going through developing knew methods in how to diagnose and navigate problems.<br />
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In this sense heuristics have dominated AI at the expense of other methods, for example diagnostic and navigation methods using neural networks and models based on insect navigation for example. The reason heuristics have been used is because computer technology has largely been focused on the development of larger machine memories, to store large amounts of heuristics and programmed commands. the heuristic and command programs are, in reality, theories written in an arcane notation, designed to be executed by a machine. The nature of the machine itself executing the file is rendered irrelevant because of the fact that the program itself is stored in a highly ordered, layered fashion in computer memory, which is a spectator not a participant in virtually all theoretical models of AI.<br />
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The only known exceptions are based on advanced forms of insect navigation or neural network simulation, which are sometimes modeled from theories based on the neurobiology of squids. However these models often come nowhere near the level of complexity seen in living neurological systems. A bee, an insect with quite sophisticated navigational intelligence, for example has around 900,000 neurons, humans have 90 billion. The largest neural networks ever created by chains of supercomputers simulate, at best, only around 1% of the human brain.<br />
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In this sense, ideas such as"the singularity" is a very misleading subject, based more on fiction than on real science. There may very well be a revolution in what is commonly labelled "AI", but unless there is going to be a severe change in diagnostic and navigation of problems in computing then the field on AI will continue to be based on group force heuristics, error correction and new techniques of object-oriented computer programming, such that it could at best generate a so-called "oracle", but not a "genies" or "sovereign" machine that some of the literature predicts.<br />
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It is important to understand that by machine intelligence we are always, if we are being honest, talking about the structure of a computer program, as the operations will always be a series of instructions of what the program can to do and what lower level programs need to be selected to complete the circuitry generated by the program firmware, which only then influences the hardware. Therefore we actually need a theoretical basis to generate intelligent programming, which requires new techniques in heuristics, self correction and new forms of object-oriented programming which, in my opinion, has not been developed. Simply looking at a simple Moore's law plot is not enough, even ignoring the fact that it is beginning to level off with the limits of silicon being reached.<br />
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The Turing Test is the most commonly cited method of testing how intelligent a machine program is by having the human user communicate with both a machine and another human via computer screens and the user being unable to judge which is the human, based on answered questions that the user asks.<br />
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Summarized in the Venn Diagram below, we might also see the Turing Test as more accurately a test of human intelligence with respect to human behavior rather than to purely test machine intelligence<br />
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<a href="http://hplusmagazine.com/sites/default/files/images/articles/feb10/turing-test2.gif"><img border="0" src="http://hplusmagazine.com/sites/default/files/images/articles/feb10/turing-test2.gif" /></a><br />
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Here we see that the computer could easily have a set of heuristics which it would use to implement responses to answers in order to generate positive results. Moreover, if trained, the computer could use metaheuristics in response to patterns of questions which, under its training exercises, it knows that will give positive results.<br />
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At such a point asking whether or not the machine is "thinking" will be meaningless - it will be obvious that it processes intelligent action based on a selected strategy, not a complex theoretical model where the computer has been programmed to contemplate.<br />
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The real barrier therefore is to have the computer then generate predictive models based on a theory of human intelligence, which is something which we are no where near completing.<br />
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However, it would be naive to expect machine intelligence to emerge based on raw computation power alone as although the mathematical model and the computational power used to solve problems will have been developed in a gradual process, the selection of which models work best will have been selected over the models that had failed, which would be promptly deemed obsolete and eliminated. It would simply make no sense for a computer program to chose a possible solution unless it was based on a predictive model that was the best one, for all time every time. The problem is we don't have the best predictive model for how intelligence arises, least of all for individual human beings.<br />
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Moreover, to have a complex model of intelligence that works for all time neglects the fact that the most complex dynamical systems are highly sensitive to slight changes in initial conditions, such in the case of a sensory data input. This is of course concerning the field of Chaos Theory. Chaos would clearly be inherent in even the most ideal predictive model of intelligence, even more so based on its complexity.</div><div dir="ltr" style="text-align: left;" trbidi="on"><br /></div><div dir="ltr" style="text-align: left;" trbidi="on">In Quantum Systems Chaos is also apparant, especially when we begin to examined the behaviour of certain states of matter which are "Chimeras" of random fluctuations and determined exchange rules.</div><div dir="ltr" style="text-align: left;" trbidi="on"><br /></div><div dir="ltr" style="text-align: left;" trbidi="on">We will be focusing on this very interesting part of the story. As we shall see, the chimera state of a quantum network is really the horse pulling the wagon with regard to solving problems.</div><div dir="ltr" style="text-align: left;" trbidi="on"> <br />
<img height="247" src="http://c2down.cyworld.co.kr/download?fid=64222746db8f5c5402e52751be3dc2a0&name=r.jpg" width="400" /><br />
<i>Complex predictive models of intelligent response, even based on the most ideal simulations, would be likely to be inherently chaotic.</i><br />
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Therefore machine intelligence, if ever created, will almost certainly be an emergent behavior of some system of heuristical problem solving strategy rather than from an elaborate mathematical model implemented by massive computation.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgJk44qB6wAUMEZgyxlago3097eeeZfMdHlzbhdpYUz2BB3PhchCQK0hLR9bpOT8ZvL_G85sj3OEpXEZsCPlQ_GOtVbbYbNEBjKby7r57G9_XR4UqRc6ZCJb6gKfoOhRIB32aRlIfHYla85/s1600/brains+left+side+graphic.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgJk44qB6wAUMEZgyxlago3097eeeZfMdHlzbhdpYUz2BB3PhchCQK0hLR9bpOT8ZvL_G85sj3OEpXEZsCPlQ_GOtVbbYbNEBjKby7r57G9_XR4UqRc6ZCJb6gKfoOhRIB32aRlIfHYla85/s1600/brains+left+side+graphic.jpg" /></a></div>
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With machine intelligence as an emergent behavior of problem solving, the program heuristics will be written based on a model, however the neural network training or global optimization model will refine the process of what heuristic is chosen, in a fashion more similar to biological evolution, as it will retain the vestiges of the failed attempts at solving the problem and will actually know not to select them. The more attempts it makes, the more refinements and the more intelligent it becomes. An intelligent program, in this view, will be literally based on the theory of, as Samuel Beckett says "Ever tried. Ever failed. No matter. Try Again. Fail again. Fail better." Such a model would require neural networks to train.<br />
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Also, it is an often case in engineering, either in experimental or theoretical (and thus programming) terms, that a design that has evolved a use in a living organism is completely useless for a machine. A bird's wing for example requires an intricate flapping motion to achieve lift, with muscles and tendons needing a constant blood supply to expand and contract and to heal any ligament damage that may be caused. An aircraft wing for example is much more simpler in design and depends on the jets or propellers to create an air flow to provide lift, which is more efficient than a constant flapping motion and provides a much more stable flight for larger craft. Increased efficiency and decreased variation in artificial selection of machines is a very general rule. The initial designs of aircraft were incredibly varied, even going back to the early designs of Leonardo Da Vinchi, but when the best design of aircraft wings were invented the variation was not simply diluted but largely eliminated. Improvements were made based on a single design, in a sense a single common ancestor, which only began to diversify due to aircraft engineering being pushed into different extremes, supersonic speeds, space transport ect.<br />
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In essence a designed object may function the same as an evolved one, in this case to produce flight, but efficiency compels a designer to modify the designs which may eventually function entirely different from their natural counter parts.<br />
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Likewise with intelligent machines, intelligence in a computer system may be engineered in an entirely different way to intelligence that arises in biological systems, such as in insects which can display quite complex systems of organised intelligence in the case of ants, termites and bees. In this sense, machine intelligence may be as different from biological intelligence as the wing of a bee differs from the wing of a B-22 Bomber.<br />
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In that case, we would have to ask: "is the machine displaying a comparison to biological intelligence just because it appears intelligent?" But it turns out, even taking only a small step of logic, that this as meaningful a question as asking "does an aircraft fly just because it appears to fly?". If we look at both an airplane and a bee fly, do we describe them both as flying? The answer, in almost all cases, is obviously yes, so much so that the question becomes meaningless. Therefore a machine, although mimicking a biological function, for all intents and purposes, is carrying out that function.<br />
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In this case we may continuously fail again and again at creating a programming routine which completely mimics intelligence, i.e. something which cannot always pass the famous Turing Test. Worse, we may fail to recognize that we have created a form of intelligence or even consciousness in a computer at all, before it begins to behave in a conscious way, ways which may be very subtle. We do not yet know much about how intelligence can emerge, even in existing intelligent organisms.<br />
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Our only clue to building an intelligent computer program is to see how programs could use heuristics to solve problems. Therefore by developing various heuristics and operations for their use, the first test for machine intelligence would be to measure how efficient computers are at using the encoded heuristics negotiating the sensory information being provided to it with the speed or dexterity of its reactions.<br />
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Chess game playing computer programs do this, with every possible heuristic strategy being encoded by the human programmers and the program difficulty being measured by how efficient the program uses heuristics to negotiate the input information from the opponent with the response in order to win the game.<br />
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<i>IBM's Deep Blue Chess Playing Computer. Instead of focusing on the calculations and permutations involved in chess, computers like Deep Blue just need to rely on a memory bank of pre-programmed strategies, which they use as heuristics.</i><br />
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With metaheurisitcal machine learning, a computer can be trained, with neural networks for example, to select particular lower level heuristics if certain patterns are identified that it had seen before in its training exercises. This increases the probability of selecting strategies which can win the game quickly. However, since the intrinsic nature of heuristics is still there, the computer can fail.<br />
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In more elaborate programs, still using metaheuristics negotiations between the sensory patterns identified and a theoretical global optimization model can be used make predictions, process them into decisions and then negotiate these decisions with an implemented response to stimuli.<br />
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Since the input sensory patterns is learned behaviour, we can then see that the jump to hyperheuristics would be to impliment advanced global optimization models. These could be based on Monte Carlo probabilistic or on quantum computer algorithms.<br />
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In the case of quantum computers, we have seen that the Adiabatic Quantum Computer, AQC, algorithms can utilize lower heuristics to replace the complex procedure of having a initial Hamiltonian having a uniform superposition as ground state.<br />
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The ability to start the quantum evolution with a state which is a guess to the solution of the problem, such that the initial guess corresponds to the non-degenerate ground state of the initial Hamiltonian, as it is required by AQC. The protocol from the initial Hamiltonian would then be run as a reference to run another protocol, using another Hamiltonian, twice as fast. This would mean, ideally, that we would need 2 or more Quantum Computer processors would be necessary to implement the algorithm in full, or we can cheat by having a probabilistic selection of 2 separate adiabatic protocols.<br />
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It is also import to note that the use of a single quantum computer allows the use of special "quantum heuristics" which refines the outcome of the near-adiabatic quantum evolution for further use of the protocol.<br />
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Therefore self-adapting optimization models can be created using adiabatic quantum computing, by combining classical heuristics of possible predicted solutions to the the Hamiltonian final state with the quantum heuristics which refine the actual Hamiltonian final state and using these refinements in a feedback for further use of the protocol.</div>
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It is therefore not inconceivable that self-adapting optimization models that work based on quantum protocols could be used in training exercises, with the known classical prediction heuristics being selected based on patterns of refinement observed on quantum heuristics. Moreover the predictive models themselves could be developed to be refined based on the compared refinements implemented through quantum heuristics which would create a highly sophisticated means of computer adaptation. At this point we can only speculate at what emergent properties could arise from these protocols.<br />
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One technique which will be of use of self adapting quantum protocols will be to use adaptive techniques to replace the brute-force methods of pattern recognition, translation and search techniques.<br />
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Pattern recognition programs are currently brute force, based on the computation of geometrical rules and best fitting them to observed data rather than using heuristics, such as neural networks, to train multilayered commands with lower level pattern recognition strategies and then implement the results of such training exercises on observed data.<br />
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<a href="http://vv.carleton.ca/~neil/neural/neuron9.gif" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img alt="[Competitive learning scheme]" border="0" src="http://vv.carleton.ca/~neil/neural/neuron9.gif" /></a>In its simplest form, pattern recognition uses one analog neuron for each pattern to be recognized. All the neurons share the entire input space. Assume that the two neuron network to the right has been trained to recognize light and dark. The 2x2 grid of pixels forms the input layer. If answer #1 is the `light' output, then all of its dendrites would be excitatory (if a pixel is on, then increase the neuron's score, otherwise do nothing). By the same token, all of answer #2's dendrites would be inhibitory (if a pixel is off then increase the neuron's score, otherwise do nothing). This simply boils down to a count of how many pixels are on and how many are off. To determine if it is light or dark, pick the answer neuron with the highest score.<br />
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<img align="left" alt="[Neural weightings for letter `A']" height="253" src="http://vv.carleton.ca/~neil/neural/neuron10.gif" width="121" />The example above is so simple that it is in fact a complete waste of a neural net, but it does demonstrate the principle. The output neurons of this type of network do not require thresholds because what matters is highest output. A more useful example would be a 5x7 pixel grid that could recognize letters. One could have 26 neurons that all share the same 35 pixel input space. Each neuron would compute the probability of the inputs matching its character. The grid on the left is configured to output the probability that the input is the letter <b>A</b>. Each tile in the grid represents a link to the <b>A</b> neuron.<br />
<img align="right" alt="[Network that recognizes digits from 0 to 9]" height="341" src="http://vv.carleton.ca/~neil/neural/neuron11.gif" width="343" />Training these pattern recognition networks is simple. Draw the desired pattern and select the neuron that should learn that pattern. For each active pixel, add one to the weight of the link between the pixel and the neuron in training. Subtract one from the weight of each link between an inactive pixel and the neuron in training. To avoid ingraining a pattern beyond all hope of modification, it is wise to set a limit on the absolute weights. The Recog character recognition program uses this technique to learn handwriting. It is a Windows application that can quickly learn to distinguish a dozen or so symbols. The network to the right was created by Recog to recognize the digits from 0-9.<br />
A more sophisticated method of pattern recognition would involve several neural nets working in parallel, each looking for a particular feature such as "horizontal line at the top", or "enclosed space near the bottom". The results of these feature detectors would then be fed into another net that would match the best pattern. This is closer to the way humans recognize patterns. The drawback is the complexity of the learning scheme. The average child takes a year to learn the alpha-numeric system to competence.<br />
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Computational translation of language is also a brute force search of the lexical semantics of the known meanings of words, in a sense looking up an already ordered database of words i.e. a thesaurus or dictionary. These are low level heuristics, rules and definitions which can be broken down to axioms. Aspects like grammar must also come from information contained in low level heuristics in the same sense that the meaning of words have to. These ordered search models can be used as low level heuristics in multilateral neural networks.<br />
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Low level alphabetical search being used as a low level heuristic in a neural network model for translation and/or learning of language. Such self-organizing maps can also be used to simulate training regimes for how words can be distributed in a database to be used to test multilayer neural network programs to recognise patterns in an unknown database.<br />
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The model presents a “semantic network” and a “lexical network.” The semantic network consists of nine distinct Feature Areas (upper shadow squares), each composed of 20 × 20 neural oscillators. Each oscillator is connected with other oscillators in the same area via lateral excitatory and inhibitory intra-area synapses, and with oscillators in different areas via excitatory inter-area synapses. The lexical area consists of 20 × 20 elements (lower shadow square), whose activity is described via a sigmoidal relationship.<br />
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The two main activation functions used in current applications are both sigmoids, and are described by</div>
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<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19.2px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="\phi(v_i) = \tanh(v_i) ~~ \textrm{and} ~~ \phi(v_i) = (1+e^{-v_i})^{-1}" class="tex" src="http://upload.wikimedia.org/math/8/e/9/8e997013f9e741f7e4d603a39766bc5e.png" style="border: none; vertical-align: middle;" /></dd></dl>
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Moreover, elements of the feature and lexical networks are linked via recurrent synapses (WF, WL). - See<br />
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The main goal of these models is to develop a computational model of the mechanisms involved in learning to read. The model is constrained by data showing the use of a specific orthographic code for words, training regimes based on the distribution of words in databases and unsupervised learning algorithms<br />
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and a biologically plausible neuro-computational model in order to represent the implicit part of learning to read. The simulation results are compared to performance in children. Our model, a self-organizing map, successfully simulates the behavioral results. This adds support to the hypothesis of unsupervised learning of orthographic word forms in reading acquisition.<br />
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In these models here is only attention given to the lexical semantics of words in these models with no attention to the logical semantics of words, which would have to use metaheuristical global optimization to form a presupposition of meaning. There is of course a reason for this, as a typical computer program would crash due to the sheer number of loops that would arise from revisions of the chosen heuristics needed for the translation.<br />
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We can see therefore that the most common methods of handwriting recognition must be a combination of the brute force methods of pattern recognition which use a library of convolution neural networks, as well as the brute force translation method that uses a library of ordered search matches.<br />
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A Convolution Neural Network (CNN) is really high to small character classes such as digits or English alphabet (26 characters). However, creating a larger neural network that can recognize reliably a bigger collection (62 characters) is still a challenge. Finding an optimized and large enough network becomes more difficult, training network by large input patterns takes much longer time. Convergent speech of the network is slower and especially, the accuracy rate is significant decrease because bigger bad written characters, similar and confusable characters etc.<o:p> </o:p></div>
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The proposed solution to the above problems is taking place of a unique complex neural network by multiple smaller networks which have high recognition rate to these own output sets. Each component network has an additional unknown output (unknown character) beside the official output sets (digit, letters…). It means that if the input pattern is not recognized as a character of official outputs it will be understand as an unknown character.</div>
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Google search does use metaheuristics to refine the final matrix of initial search requests, however its translation feature is clearly lacking in the use of heuristics, using statistical network means instead, to cross the non-trivial implementation of protocols in programming that uses translation by logical semantics.<br />
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Therefore the development of global optimization models that use self-adapting heuristics, such as in the case of Adiabatic Quantum Computation could be the key to computers performing accurate face recognition as well as language learning and translation that would spawn a plethora of new software technology that could do tasks that classical computers struggle with and ultimately beat them at recognizing patterns.<br />
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This is the key to having computers that solve problems normally associated with human intelligence. To have any strategy that recognizes patterns and makes adaptive and predictive modifications to the strategy based on those patterns in order to solve a particular problem will be the only triumph in machine intelligence needed to create huge progress in finding errors in computer code to recognizing human faces as well as implementing response measures accordingly.<br />
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Despite the sophisticated mimicry that these machines may display, it may be a sobering conclusion that the ability to create a machine that truly mimics all of the functions of a biological organism, from its mechanics to its neurological processes, to the point that we have created a true copy of a biological system may very well be beyond contemporary science. We may very well be surrounded by machines with a high level of coordination that mimics life, but the complexity involved in copying every function of a living organism with a series of non-biological analogues may be an impossible task.<br />
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Moreover, the rule that we have imposed on this task, namely using entirely non-biological structures to mimic biological structures may not only be a completely arbitrary rule, it could be a rudimentary one. After all, for a human to recognize language or a pattern it does not need a network of superconducting quantum computers running on vast amounts of power in a massive facility, all it needs is a large, cellular based, neural network, working at a temperature system relatively close to its external temperature range running inside a biological object that is mostly a mass of fat and liquid, namely the human brain, that only needs the energy of 20 Watts. Compared to that, a quantum computer, despite the technological complexity, is still a rather rudimentary object when compared with biological structures, with no way of competing with a human at the very least in terms of energy efficiency and not to mention that the human brain is a self-organisational system that not only is self learning but self-assembling.<br />
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Who benefits from a quantum computer?</h2>
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<a href="https://blogger.googleusercontent.com/img/proxy/AVvXsEi3oFmBxeoVuVciD6735-zwyoZNnw9WWWtIjoawlodIN3RQ1IAoLUiaMmKF73cs6i49g95vmXGkV4tmVSkJL2dTKjkW22a9Tq5SW_gliSMtkffW4yLxnh1oAjaoGOFqWgdhX9y6lJ823r1ofz1TsYoTrCuE3ZfQbWN25J8yny8lfkulAZfxusdOmq1ywQ=" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" height="297" src="http://singularityuniversitynl.files.wordpress.com/2012/04/quantum-dwave.png" width="400" /></a></div>
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As Richard Feynman appropriately stated "quantum mechanics behaves like nothing you have seen before" and similarly with quantum computers we see applications which have never been seen before and therefore there are questions as to whether or not they will benefit everyday people or will remain as the tools of research.</div>
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Transistors, first developed by Bell Labs with AT&T Corporation subsidies, were largely used for research/development and space/military, at prototyping institutions such as Lincoln Labs at MIT before their incorporation into independent industry, in companies such as Dell and IBM.</div>
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Similarly, Lasers were, since their creation in the 1960's, largely used for research 1960's-70's, then space/military/industry 1970's-80's, then in consumer electronics 1980's-90's. The most powerful lasers are still used largely by the first 2 private sectors with these developments trickling down into the public sector. For quantum computers a similar trend is certainly going to be the same, and we are already witnessing the transition between research and development and space/military with quantum computers, with NASA, Lockheed-Martin and Google all owning a quantum computer.<br />
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It is also rumored that The National Security Agency, NSA, also wants a quantum computer within the decade of 2020, almost certainly function in some capacity with its Utah Data Center outside Salt Lake City which will undoubtedly increase its surveillance powers.<br />
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In effect, the use of this technology, in government and corporate capacity, could be 30 years away at least. Using a quantum computer, Shor's algorithm could be used to break public-key cryptography schemes such as the widely used RSA scheme. RSA is based on the assumption that factoring large numbers is computationally infeasible. So far as is known, this assumption is valid for classical (non-quantum) computers; no classical algorithm is known that can factor in polynomial time. However, Shor's algorithm shows that factoring is efficient on an ideal quantum computer, so it may be feasible to defeat RSA by constructing a large quantum computer.<br />
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" /><br />
This will mean it will be possible to break into what were once previously considered to be secure communications, i.e. classical communications that use classical encryption protocols that will not have upgraded to a quantum key encryption standard, which will be impossible to break, even for quantum computers.(My blog will have another article based on this research soon)<br />
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The ability to develop emergent behavior and enhanced metaheuristic search is something which is about to reach a revolutionary point so the ability to produce very highly coordinated robots, that navigate in the same fashion as insects, along with metaheuristic search engines and code compilers that will be the beginning of self correcting, self-organizing, perhaps even self assembling machines will be coming our way over the next 30 years.<br />
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Probably the most visible aspect of this may be fully automated transport systems, robot cars, planes, aircraft, ect that will almost certainly spin off from the immense state military funding these technologies get for drones, robot tanks, robot ship steering and neural networks for shape/image recognition of inanimate and animate objects.<br />
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Therefore it seems to be big business, megacorporations, military and intelligence agencies which benefit most from the development and operation of quantum computer technology, which explains government subsidies for their development in the state sector, often under the guise of defense but in reality for generating new sectors of the economy which will become profitable in a future market niche after a sufficient demand is created.</div>
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It is hoped that these developments will help create a practical computer system commanding airborne drones, aircraft or space vehicles, making them less chaotic and prone to causing unpredicted damage to themselves and to humans. As machines become more complicated, in particular deep space probes, failure is much more likely. The massive amount of programming that goes into robotic spacecraft requires that several years are spent checking for all possible errors in the code and running simulations on powerful computers to actively find errors.<br />
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The New Horizon's Spacecraft is a perfect example of this. Before launch, huge teams of engineers were required to primarily check the computer software for errors as well along with performing controlled experiments, exposing the probe to artificial vacuum, radiation, temperature and vibration extremes, monitoring the sensor data and checking for any problems the probe computers has with receiving the information and relaying it back to earth or with receiving information from earth and responding accordingly.<br />
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This is of course standard with any spacecraft, but as one of the fastest spacecraft ever created, second only to Voyager 1, the spacecraft will only have less than a day to carry out its full mission on Pluto as it will be moving so fast that the gravitational interaction with the planet will be negligible. To prepare for the sheer speed of the encounter, the team at NASA had to do a trial run of the interaction when the spacecraft had a flyby of Jupiter and is continuing simulations to try and perfect the Pluto flyby ever since.<br />
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The Mars Curiosity Rover also had similar trials, particularly on the speed at which the probe descended to the surface and how sensitive the robotic equipment was to vibrations. A single line of code which prevented the robotic crane commands from working properly in a decent would have meant automatic failure. By far the Curiosity Rover landing was the most sophisticated ever attempted so far on Mars.<br />
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It is therefore critical to advance the idea of using quantum computers and metaheuristics to check for errors, replacing less sophisticated code searching techniques, in same sense as the development of supercomputers being critical in the 1990's and 2000's for the replacement of less sophisticated, and altogether more expensive, wind tunnel experiments for aircraft and testing of nuclear fission and fusion.<br />
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The use of quantum computers in predicting errors and providing new ways to perform simulations, based on the idea of metaheuristics, could be as important to our survival in the next few decades as the first use of supercomputers to predict the weather. As developments increase, their use in the public sector will become much more apparent. design of new drugs and pharmaceuticals using models of protein folding in quantum computers as well as more accurate financials and weather predictions will mean that the economy will benefit from the investment in the long run.<br />
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References<br />
[1] Richard P. Feynman. Simulating Physics with Computers. International Journal of Theoretical Physics, 21 (6/7) pp. 467‐488 (1982)y The Feynman Lectures on Computation. Penguin Books (1999). [2] K. Grover. A fast quantum mechanical algorithm for database search. Proceedings 28th annual ACM SymposiumTheoryofComputingpp212–219(1996) Symposium Theory of Computing, pp. 212219 (1996). [3] P. Shor. Polynomial‐Time Algorithms for Prime Factorization and Discrete Algorithms on a Quantum Computer. Proceedings of the 35th Annual Symposium on Foundations of Computer Science, pp. 124– 134, IEEE Computer Society Press (1994). [4] A.M. Childs, R. Cleve, E. Deotto, E. Farhi, S. Gutmann, and D. Speilman. Exponentialalgorithmicspeedup by quantum walk.Proceedings of the 35th ACM Symposium on the Theory of Computation (STOC ’03) ACM, pp. 59.68 (2003). [5]ChristianB.Anfinsen.Studiesontheprinciplesthatgovernthefoldingofproteinchains. [5] Christian B. Anfinsen. Studies on the principles that govern the folding of protein chains. Nobel Lecture, December 11, 1972. [6]Quantum Computation by Adiabatic Evolution. E. Farhi, J. Goldstone, S. Gutmann, and M. Sipser. ArXiv:quant‐ph/0001106 []dbllhlddflbl [7]A quantum adiabatic evolutionalgorithmappliedtorandom instancesof anNP‐complete problem. E. Farhi, J. Goldstone, S. Gutmann, Joshua Lapan, Andrew Lundgen, and Daniel Preda. Science vol. 292 pp. 472‐476 (2001). [8]Howpowerfulisadiabaticquantumcomputation?WinVanDam,MicheleMosca,andUmeshVazirani. [8] Howpowerfulis adiabatic quantum computation?WinVan Dam, MicheleMosca, and UmeshVazirani. Proceedings of the42ndSymposium on the Foundations of Computer Science pp. 279‐287 (2001). [9] Universal quantum walks and adiabatic algorithms by 1D Hamiltonians. B.D. Chase and A.J. Landhal. Arxiv quant‐ph/0802.1207 [10]QMhiAMih D(1999) [10] Quantum Mechanics.A. Messiah. Dover (1999).<br />
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In this video I showcase a type of light circuit I made back in 2015 (which I now call Feynman's Fireflies) to demonstrate the principle of emergent behavior in metaheuristic systems, in this case of metaheuristic synchronization of light signals in a network that has been programmed using a Discrete form of the Firefly Biomimetic Algorithm.
To my surprise, the wave like patterns that emerged from the discontinuous pas coupling of the oscillation LED flashes reminded me of the description of quantum mechanics offered by The Physicist Richard Feynman, The Path Integral Interpretation of Quantum Mechanics.
In this description, the path a quanta (i.e. a particle) such as a photon or electron follows is a sum over all possible histories in phase space where each possible path is an element of the sum given by the individual phase amplitude for a given path, eiS/h-bar (where S is the action i.e. the integral over time of the Lagrangian, L, as shown in the upper diagram; the Lagrangian of a path represents the difference between a path’s kinetic and potential energy) by Euler’s formula has a real part and a complex (or imaginary) part so requires for graphical representation can be typically done on an Argand diagram (which has a real horizontal axis and a complex or imaginary vertical axis).
In the case of our cycling LEDs they start in a state where they are unsynchronized and follow a sum over possible paths to achieve synchronization with random oscillations that are vulnerable to external stimuli and create an effect which reminded me of the "observer effect" which can happen in the path integral description of an electron following many probable pathways through a double slit forming a wave like pattern when undisturbed and a linear pattern when observed/disturbed.
I constructed a model describing this effect in more detail and I was able to find some very interesting relationships between this pseudo-quantum behavior and classical behavior of metaheuristic and coupled networks which show many equivalences which have been of interest to me for some time and which I am happy to share.</span></div><div><span color="rgba(0, 0, 0, 0.87)" face="Roboto, Noto, sans-serif" style="font-size: 15px; white-space: pre-wrap;"><br /></span></div><div><span color="rgba(0, 0, 0, 0.87)" face="Roboto, Noto, sans-serif" style="font-size: 15px; white-space: pre-wrap;">video of the making of this circuit:</span></div><div><span color="rgba(0, 0, 0, 0.87)" face="Roboto, Noto, sans-serif" style="font-size: 15px; white-space: pre-wrap;"><br /></span></div><div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen="" class="BLOG_video_class" height="266" src="https://www.youtube.com/embed/2d6dHHhKOOE" width="320" youtube-src-id="2d6dHHhKOOE"></iframe></div><br /><div><br /></div><div><span color="rgba(0, 0, 0, 0.87)" face="Roboto, Noto, sans-serif" style="font-size: 15px; white-space: pre-wrap;"><br /></span></div>MuonRayhttp://www.blogger.com/profile/03712859045968965104noreply@blogger.com0tag:blogger.com,1999:blog-5996268492156364201.post-65502479062853432972020-10-03T05:32:00.002-07:002020-10-03T09:20:51.704-07:00Image + Video Segmentation in Near-Infrared Using HSV Color Spaces with OpenCV in Python<p></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjxGehbG8C8PytuvtGxk2NembR5cTvLFX4ult6UnjlkdekRehB-nKxHYXuQCBa7I5RKx9Oc-ebs5OSh_8YV7VqI1CjP7mw4KSSdJJ_gtVztWHHXjpdK2Pw6CO3pYUTkBnWGcJfi2poZQXeD/" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="1330" data-original-width="2048" height="252" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjxGehbG8C8PytuvtGxk2NembR5cTvLFX4ult6UnjlkdekRehB-nKxHYXuQCBa7I5RKx9Oc-ebs5OSh_8YV7VqI1CjP7mw4KSSdJJ_gtVztWHHXjpdK2Pw6CO3pYUTkBnWGcJfi2poZQXeD/w470-h252/SegmentedInfraBlueNDVI.jpg" width="470" /></a></div><br /><br /><p></p><p>Here I will be sharing a technique to perform a simple kind of image segmentation used to separate certain objects visible in the near-infrared and ultraviolet using the hue, saturation and value values (HSV) contained in the color space with OpenCV in Python. This is a useful tool in the processing of NIR images and video when we want to search for vegetation in an image using a defined threshold. Moreover, we can perform fast NDVI analysis on the examined region in the video clips. </p><p>First of all we need to set up an idea of what we mean by segmentation. In this case it means to literally cut out an object that has a particular well defined colour in the image taken.</p><p>We should remember what we mean by colour itself in an image our brain, or a computer, "sees".</p><p>In a very real sense, we see with our mind's own programming, not with our eyes which merely sense changes in sensitivity across sensing elements, the cones in our eyes. For example consider sunlight shining on this apple</p><p></p><div class="separator" style="clear: both; text-align: center;"><img alt="" data-original-height="532" data-original-width="820" height="208" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi1PxIam-QhtrDOUNV3TcPYQEgYnTy5HorNiC3hXX0ZWTAWl-1VxhsbdK_V1FtIaJb6KkUAciaqY44StkXzblNvbBM3YYLoFbrkQUhr95E38h3z2lzIOHT-ZA3Wf8k032IyM0EWEYIKok6e/" width="320" /></div><br /><br /><p></p><p>What is important to remember is that we do not see the red spectrum of light, but we see every other spectrum except that one! Every color except red is absorbed by the object, our eye sees this and sends a message to our brain. Our brain then labels this this as a red apple. </p><p>This is a labeling procedure that our brains have developed with a dependence on the level of light exposed, not the intrinsic colour of the object. The best example of this dependent effect of light on color is seen when a colored lamp is put on something. If we had a red lamp and turned it on in a dark area, everything would appear to be red... that's because there's only red light to bounce off of it!</p><p><br /></p><p></p><div class="separator" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em; text-align: center;"><img alt="" data-original-height="1536" data-original-width="2048" height="240" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg4_LqPdbee1bfLgcYNJhkAFCl7NsGKsrLZPZrblUCD14nPP41GoVRKd8jNNZZqpSEbUPiDmNn9A3w_b1xY2LLg8JaL-ADwsQ7iQ4_siKNVDew3duew1mZtFNapRD5pmy6qhyphenhyphenwwuDRt9s7o/" width="320" /></div><br /><br /><p></p><p>These special red street lamps were designed to stop harmful light pollution that can confuse and disturb the health and balance of nocturnal animals, such as bats, insects and even reptiles such as lizards and sea turtles. As we can see ourselves, otherwise green plant and black road appear red even though they would not appear red when radiated in balanced proportions of red green and blue light.</p><p><br /></p><p>This principle is also true when we take photographs using a CCD sensor, which is equipped with a filter as to properly correlate with our own sense of vision, with some adjustments to sensitivity to light exposure, aperture size and shutter speed.</p><p>When the light of a scene enters the camera lens, it gets dispersed over the surface of the camera’s CCD sensor, a circuit containing millions of individual light detecting semiconductor photodiode sensors. Each photodiode measures the strength of the light striking it in Si unit called “lumens.” Each receptor on this sensor records its light value as a color pixel using a specific filter pattern, most commonly the Bayer filter.</p><p></p><div class="separator" style="clear: both; text-align: center;"><div class="separator" style="clear: both; text-align: center;"><img alt="" data-original-height="716" data-original-width="2486" height="153" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhdnIaxmPnBV9yy9rd299L7AX0T3pR26jPkcDa8vWzKyOKAC2QP0yGzv8o5thddonol2_Ori0gJ_yigSaIweMECgKcNYf_dCohvEyLknpFOgwqrbMbmds5gbxu5GrInoZb1khCiPOo524uF/w532-h153/Camera-Capture.jpg" width="532" /></div></div><p></p><div class="separator" style="clear: both; text-align: center;"><i>A real image being focused and projected onto a CCD image sensor containing a colour filter. The Bayer Filter uses color filters and direct the individual colors red, green, blue onto individual pixels on the sensor, usually in an RGGB arrangement (i.e. 2X2 [RG;GB] ).</i></div><div class="separator" style="clear: both; text-align: center;"><i><br /></i></div><p>The camera’s image processor reads the color and intensity of the light striking each photoreceptor and maps each image from those initial values in a function called an input device transform (IDT) for the digital sensor information to be used to store a reasonable facsimile of the original scene in the form of raw data containing the color channels in separate channels for processing and colour processing. </p><p>To display the image the raw data must be processed to have a color scale and luminance range to try and reproduce the color and light from the original scene. A display rendering transform (DRT) provides this mapping and are typically implemented in a form of preset ranges which are obtained using calibration standards, such as the SDR and HDR standards. Often raw data must be modified by standalone image processing software with the editor in command of controlling aspects such as contrast, gamma correction, histogram correction and so forth between the separate colour channels before we are given the final satisfactory image</p><p>With the RAW image data a process known as Demosaicing (which is also called de-Bayering in our case) is used to reconstruct a full color image, with RGB values at every pixel, from the incomplete color samples output from an image sensor overlaid with a color filter array (CFA).</p><p></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgt-i5jFPJ8n8WYNe2Y8ni98hWcls1H1BBdrudRFsXuCmR-vji0KUCG9J6vryT6DqEGTr2qVhjuYpjUOsFLPFOWowhOFXrWxoq56hMo81UnY7Rjvee5z-Y_w4b0jlEAFeEkStIyK8LvAZEc/" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="693" data-original-width="1600" height="205" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgt-i5jFPJ8n8WYNe2Y8ni98hWcls1H1BBdrudRFsXuCmR-vji0KUCG9J6vryT6DqEGTr2qVhjuYpjUOsFLPFOWowhOFXrWxoq56hMo81UnY7Rjvee5z-Y_w4b0jlEAFeEkStIyK8LvAZEc/w471-h205/image.png" width="471" /></a></div><br /><br /><p></p><p>When this bitmap of pixels gets viewed from a distance, the eye perceives the composite as a digital image. </p><p>As discussed previously on this blog and showcased on my YouTube videos, I have been imaging individual plants and sections of vegetation in the near-infrared using drones for some time now. As I have shown, the plant material reflects very vibrantly in the near-infrared in the color channel which we associate with the colour red. Therefore if I want to segment the image to separate the plants from the background, using an infrared image, I can set the segmentation around the red color channel.</p><p>The individual color channels in an image, red, green and blue, can together form a group which we call a color space. In OpenCV in Python the color space is a tuple, a vector like object in the programming language. A color is then defined as a tuple of three components. Each component can take a value between 0 and 255, where the tuple (0, 0, 0) represents black and (255, 255, 255) represents white.</p><p>RGB (Red, Green and Blue) colour space is used normally to create colors that you see on television screens, computer screens, scanners and digital cameras. RGB is often referred to as an 'additive colorspace.’ In other words, the more light there is on a screen means the brighter the image.</p><p>CMYK (Cyan, Magenta, Yellow and Black) is known as the 4-color process colors for printing. CMYK is considered a subtractive colorspace. May colour printers print with Cyan, Magenta, Yellow and Black (CMYK) ink, instead of RGB. It also produces a different color range. When printing on a 4-color printer, RGB files must be converted into CMYK color.</p><p></p><div class="separator" style="clear: both; text-align: center;"><img alt="" data-original-height="260" data-original-width="600" height="170" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj6v9IzUhcgsbku2Ot58G-0F5iYMnxzrpFTWadBw21VxCG5lBiCvzbEodM6yEIPS_sqqDMKh5cNYBV8IzqP4jFYdcYMkRG6zCQRhhYnS0ZCb7zbKJw7aXkRpgDKz-38ouyztA2FAjNVEX1c/w391-h170/rgb-vs-cmyk.jpg" width="391" /></div><div class="separator" style="clear: both; text-align: center;"><i>The RGB additive colour space vs the CMYK Subtractive Colour Space. As you may notice, there are certain RGB colors you see on your computer screen (or camera) that are unable to be duplicated with basic CMYK. Also notice that CMYK Colour can appear somewhat deeper than RGB combinations. This is part of the reason they are favored in printing.</i></div><br /><p></p><p>The HSV (Hue, Saturation, Value) model, also called HSB (Hue, Saturation, Brightness), defines a color space in terms of three constituent components:</p><p><b><span style="color: #cc0000;">Hue</span></b>, the color type (such as red, blue, or yellow): Ranges from 0-360 (but normalized to 0-100% in some applications)</p><p><b><span style="color: #2b00fe;">Saturation</span></b>, the "vibrancy" of the color: Ranges from 0-100%</p><p><b><span style="color: #ffa400;">Value</span></b>, the brightness of the color: Ranges from 0-100%</p><p>Also sometimes called the "purity" by analogy to the colorimetric quantities excitation purity and colorimeric purity</p><p>The lower the saturation of a color, the more "grayness" is present and the more faded the color will appear, thus useful to define desaturation as the qualitative inverse of saturation</p><p>The HSV model was created in 1978 by Alvy Ray Smith. It is a nonlinear transformation of the RGB color space, and may be used in color progressions.</p><p>In HSV space, the red to orange color of plants in NIR are much more localized and visually separable.</p><p></p><div class="separator" style="clear: both; text-align: center;"><img alt="" data-original-height="589" data-original-width="605" height="323" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjKRU7Pg7SUDcZOxNTPYNp9M6nsY-lQ1AhCTAPZYnvEFs1mcmvg8ozaTYW3hbtAR45G5MmaCJ0cesLU00lISUh0ieJ7xUymnjbF1-vNXBgrU1b3eW4PjwZUAZ6LjZRdbpoJdXwV-mPqtmxg/w332-h323/image.png" width="332" /></div><div class="separator" style="clear: both; text-align: center;"><i>Raw image of the HSV color space, stereographic projection of the surface on a circle</i></div><br />The HSV model is commonly used in computer graphics applications. In various application contexts, a user must choose a color to be applied to a particular graphical element. When used in this way, the HSV color wheel is often used. In it, the hue is represented by a circular region; a separate triangular region may be used to represent saturation and value. Typically, the vertical axis of the triangle indicates saturation, while the horizontal axis corresponds to value. In this way, a color can be chosen by first picking the hue from the circular region, then selecting the desired saturation and value from the triangular region.<div><br /></div><div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg_36L5NeOglpphPbEwec4qURC0BZoK7kUHTA_msPtAazdwxTVphBALOp2HG2IyBn2Zm8sJHthaS5lyuQk_k45FDsNNq6kA9uj-IdK1pYBQLj9Z5PKnUugbC9CZqqlsnbnd8ZP1uoe404v6/" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="160" data-original-width="160" height="311" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg_36L5NeOglpphPbEwec4qURC0BZoK7kUHTA_msPtAazdwxTVphBALOp2HG2IyBn2Zm8sJHthaS5lyuQk_k45FDsNNq6kA9uj-IdK1pYBQLj9Z5PKnUugbC9CZqqlsnbnd8ZP1uoe404v6/w311-h311/Hsv_sample.png" width="311" /></a></div><br /><br /><div><br /></div><div><div>Graphics artists sometimes prefer to use the HSV color model over alternative models such as RGB or CMYK, because of its similarities to the way humans tend to perceive color. RGB and CMYK are additive and subtractive models, respectively, defining color in terms of the combination of primaries, whereas HSV encapsulates information about a color in terms that are more familiar to humans: What color is it? How vibrant is it? How light or dark is it? The HSL color space is similar and arguably even better than HSV in this respect.</div><div><br /></div><div>The HSV tristimulus space does not technically support a one-to-one mapping to physical power spectra as measured in radiometry. Thus it is not generally advisable to try to make direct comparisons between HSV coordinates and physical light properties such as wavelength or amplitude. However, if physical intuitions are indispensable, it is possible to translate HSV coordinates into pseudo-physical properties using the psychophysical terminology of colorimetry as follows:</div><div><br /></div><div>Hue specifies the dominant wavelength of the color, except in the range between red and indigo (somewhere between 240 and 360 degrees) where the Hue denotes a position along the line of pure purples</div><div><br /></div><div>If the hue perception were recreated, actually using a monochromatic, pure spectral color at the dominant wavelength, the desaturation would be roughly analogous to an applied frequency spread around the dominant wavelength or alternatively the amount of equal-power (i.e. white) light added to the pure spectral color.</div><div><br /></div><div>The value is roughly analogous to the total power of the spectrum, or the maximum amplitude of the light waveform. However, it should be obvious from the equations below that value is actually closer to the power of the greatest spectral component (the statistical mode, not the cumulative power across the distribution.)</div><p></p><p>The saturation and value of the oranges do vary, but they are mostly located within a small range along the hue axis.This is the key point that can be leveraged for segmentation.</p><p>Segmentation of true color images in HSV color space can be applied to Near-Infrared photography in the imaging of the near-infrared reflectance of plants. </p><p></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjVXaV4u0pluLUHft7os8D7t-thsQky9gujMeS5d7O5gfUcenQAqW81K6hbDtPS_wNH_e7tdPGhpeIObFv3GAn07AVU1hnDojiKn1Gye5FpYKYBJdve84xocNffVO2q04WOsyu6stCu6LuV/" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="3078" data-original-width="5472" height="252" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjVXaV4u0pluLUHft7os8D7t-thsQky9gujMeS5d7O5gfUcenQAqW81K6hbDtPS_wNH_e7tdPGhpeIObFv3GAn07AVU1hnDojiKn1Gye5FpYKYBJdve84xocNffVO2q04WOsyu6stCu6LuV/w448-h252/DJI_0860.JPG" width="448" /></a></div><br /><p></p><p>By applying a threshold in the interpreted red band we can separate the vegetation, taken in the NIR, from the background. </p><p>#threshold vegetation using the red to orange colors of the vegetation in the NIR</p><p>low_red = np.array([160, 105, 84])</p><p>high_red = np.array([179, 255, 255])</p><p></p><div class="separator" style="clear: both; text-align: center;"><img alt="" data-original-height="223" data-original-width="384" height="328" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjgg9ekD5ZzSkShtC8dbzMLFuxq2GsapX2IAQV9XQ4gYlW1D1cHIP_U9iDpQaQdVThxsffyzaEeWCZAIFuKDg-2FtJrYXXTiVOMfF4o-RUUy69VGOtdfo-1if1sbmqx-xXSTF2iIgBwKWwK/w563-h328/Near_Infrared_Image_Segmentation.png" width="563" /></div><br /><p></p><p>This is a fairly rough but effective segmentation of the Vegetation, in NIR, in HSV color space.</p><p>Going further we can process the selected region to form an NDVI in the region of interest. consider the NDVI taken broadly over the entire image:</p><p></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEicR4gdKe4e5mT8qUZGWNJ1UwwvPxB4l1KqBZEgff-A244O6TFW6ramuQqhCN5GG1VDmm1VGb2-BbSa4L87gFyTX57NE_HuEa-K-SamTni9QFGx4UQ933R_2hyooZayBTWfWeVBKGsqI4X0/" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="1330" data-original-width="2048" height="294" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEicR4gdKe4e5mT8qUZGWNJ1UwwvPxB4l1KqBZEgff-A244O6TFW6ramuQqhCN5GG1VDmm1VGb2-BbSa4L87gFyTX57NE_HuEa-K-SamTni9QFGx4UQ933R_2hyooZayBTWfWeVBKGsqI4X0/w452-h294/InfraBlueNDVI6.jpg" width="452" /></a></div><br /><br /><p></p><p>Processing this image takes time, especially in RAW format, and in areas where there is more concrete and rock than vegetation much of the processing seems redundant as we usually do not care about NDVI scaling of rock.</p><p>Processing NDVI in the threshold region of interest meanwhile saves a lot of processing time and creates a more precise examination of NDVI with respect to the color key:</p><p></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhw7qna1IDy6PnX_9EkjWw_mWLO3oPfHx8swTExSzOgQOzCVuMgex2Lz4TgxqF5GvsbN85huG10iMgEwDWt14E14Zg5SFqsYIFj7rivFrqfo13V-j3kUcg7-ObUnekP6h_k3lgrlVOSl9a7/" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="1330" data-original-width="2048" height="282" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhw7qna1IDy6PnX_9EkjWw_mWLO3oPfHx8swTExSzOgQOzCVuMgex2Lz4TgxqF5GvsbN85huG10iMgEwDWt14E14Zg5SFqsYIFj7rivFrqfo13V-j3kUcg7-ObUnekP6h_k3lgrlVOSl9a7/w517-h282/SegmentedInfraBlueNDVI.jpg" width="517" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><br /></div><p><span color="rgba(0, 0, 0, 0.87)" face="Roboto, Noto, sans-serif" style="background-color: white; font-size: 15px; white-space: pre-wrap;">Here we see a video showcasing image segmentation for NDVI image processing using OpenCV in Python applied on Near-Infrared drone images and video. </span></p><br /><br /><p></p><div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen="" class="BLOG_video_class" height="266" src="https://www.youtube.com/embed/LR37KuKt5nQ" width="320" youtube-src-id="LR37KuKt5nQ"></iframe></div><br /><p><span color="rgba(0, 0, 0, 0.87)" face="Roboto, Noto, sans-serif" style="background-color: white; font-size: 15px; white-space: pre-wrap;"><br /></span></p><p><span color="rgba(0, 0, 0, 0.87)" face="Roboto, Noto, sans-serif" style="background-color: white; font-size: 15px; white-space: pre-wrap;">This procedure is very useful for accurate scaling of NDVI in the region of interest, the vegetation of the image, and removing the background so as to focus on the NDVI of plant material only. </span><span color="rgba(0, 0, 0, 0.87)" face="Roboto, Noto, sans-serif" style="background-color: white; font-size: 15px; white-space: pre-wrap;">
</span><span color="rgba(0, 0, 0, 0.87)" face="Roboto, Noto, sans-serif" style="background-color: white; font-size: 15px; white-space: pre-wrap;">
</span><span color="rgba(0, 0, 0, 0.87)" face="Roboto, Noto, sans-serif" style="background-color: white; font-size: 15px; white-space: pre-wrap;">The potential for detecting plants in an image or video is also a field of interest for us in plant exploration and detection in desert regions and for plant counting by techniques such as ring detection. It is hoped these developments will be useful in detecting rare or well hidden plant specimens in remote areas for plant population monitoring in deserts and mountains.</span></p><p><br /></p><p>Notes:</p><p>OpenCV by default reads images in BGR format, so you’ll notice that it looks like the blue and red channels have been mixed up. You can use the cvtColor(image, flag) and the flag we looked at above to fix this:</p><p><br /></p><p>Python Source Codes available on the following GitHub repository:</p><p><span class="style-scope yt-formatted-string" dir="auto" style="background: rgb(249, 249, 249); border: 0px; color: #030303; font-family: Roboto, Arial, sans-serif; font-size: 14px; margin: 0px; padding: 0px; white-space: pre-wrap;">Coding available here: </span><a class="yt-simple-endpoint style-scope yt-formatted-string" dir="auto" href="https://www.youtube.com/redirect?redir_token=QUFFLUhqbEo1cEtJLW1rUmducWV0OC1ZbUN0ZFNKYklMQXxBQ3Jtc0tuNlhob204aDdZd2Y2Vjd2MERWekttUGtBSTdNNzNFVnFxQkZQZDZWYVhCMjBlUUUzYW1MaGNPNk1TbUZ2elN2eDlwVFpvOWxDcFVlWlVrVGpVT2hEY00wQWs5RzdHLVFwUmcxX3RuR3pqNjlhRXlFQQ%3D%3D&event=video_description&v=LR37KuKt5nQ&q=https%3A%2F%2Fgithub.com%2FMuonRay%2FImage-VideoSegmentationinNIRforPlantDetection" rel="nofollow" spellcheck="false" style="background-color: #f9f9f9; cursor: pointer; display: inline-block; font-family: Roboto, Arial, sans-serif; font-size: 14px; text-decoration: var(--yt-endpoint-text-regular-decoration, none); white-space: pre-wrap;" target="_blank">https://github.com/MuonRay/Image-Vide...</a><span class="style-scope yt-formatted-string" dir="auto" style="background: rgb(249, 249, 249); border: 0px; color: #030303; font-family: Roboto, Arial, sans-serif; font-size: 14px; margin: 0px; padding: 0px; white-space: pre-wrap;">
</span></p></div></div><div><br /></div>MuonRayhttp://www.blogger.com/profile/03712859045968965104noreply@blogger.com0tag:blogger.com,1999:blog-5996268492156364201.post-64110985349185807212020-09-14T10:32:00.004-07:002020-09-16T13:54:36.168-07:00New Developments in Multi-Spectral Drone Imaging in the Ultraviolet Band<div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen="" class="BLOG_video_class" height="266" src="https://www.youtube.com/embed/VgNXx0W9z1U" width="320" youtube-src-id="VgNXx0W9z1U"></iframe></div><br /><div dir="ltr" style="text-align: left;" trbidi="on"><br /></div><div dir="ltr" style="text-align: left;" trbidi="on">
Here, I report a brand new filter-based modification of a 4K Camera, The Hasselblad 20MP Camera onboard the DJI Mavic 2 Pro, to develop an ultraviolet (UV) imaging system for remote sensing.<br />
<br />
This was achieved via testing and adapting new quartz-based Ultraviolet imaging filters as well as thin film solar-filters in conjunction with commercial cameras modified using "hot-mirror" filter removal.<br />
<br />
The Hasselblad cameras used in the Mavic 2 Pro contain one of the best passive imaging,<br />
complementary metal-oxide semiconductor (CMOS) sensor, A 1-inch sensor with 20 megapixels which can be set to image using very high exposure to record in the Near-UV.<br />
<br />
The utility of these devices is demonstrable for applications at wavelengths as low as 310 nm,<br />
in particular for sensing vegetation in this spectral region. For this a novel UV-based remote sensing<br />
classification index has been developed for use in experimental Ultraviolet aerial imaging using<br />
drones.<br />
<br />
Given the relatively very low cost of these units as compared with other cameras in this field of imaging, and the fact they are integrated on a superb platform for deployment, a semi-autonomous aerial vehicle, they are suitable for widespread proliferation in the field of environmental monitoring<br />
in a variety of UV imaging applications, e.g., in atmospheric science, vulcanology, oceanography, forensics, monitoring of industry and utility structures (in particular powerlines and smokestacks), fluorescent tracer measurements and general surface measurements.</div><div dir="ltr" style="text-align: left;" trbidi="on"><br /></div><div dir="ltr" style="text-align: left;" trbidi="on">I am beginning experimental testing of this technology in Gran Canaria over the next few months. I have already begun to construct test image datasets for analysis using a prototype Normalized UV Absorption Index (NUVAI)</div><div dir="ltr" style="text-align: left;" trbidi="on"><br /></div><div dir="ltr" style="text-align: left;" trbidi="on">With this index I hope to be able to classify vegetation and surface features based on their UV absorption characteristics and compare with the NDVI taken using the same camera with my already extensively tested Infrared Filters.</div><div dir="ltr" style="text-align: left;" trbidi="on"><br /></div><div dir="ltr" style="text-align: left;" trbidi="on"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhG_zloZQal8ZYnrwvI2Uy78iz9uCJnE7stzWeu0Lvpu5tHcMf8nKwRuomfwR4CkAucJd72YPGHGaWwk_TfGiH5LxKsV-ir4Gfz8t2IIW11Iza2XuRLZI8m2Ajk5I1tYpcAc1Ls6a-W_vfa/s5472/DJI_0880.JPG" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="3078" data-original-width="5472" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhG_zloZQal8ZYnrwvI2Uy78iz9uCJnE7stzWeu0Lvpu5tHcMf8nKwRuomfwR4CkAucJd72YPGHGaWwk_TfGiH5LxKsV-ir4Gfz8t2IIW11Iza2XuRLZI8m2Ajk5I1tYpcAc1Ls6a-W_vfa/s320/DJI_0880.JPG" width="320" /></a></div><div class="separator" style="clear: both; text-align: center;">Ultraviolet Drone Aerial Image</div><div class="separator" style="clear: both; text-align: center;"><span style="text-align: left;"><br /></span></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><span style="text-align: left;">Using Python coding I have digitally processed some of the test images already and hope to perform similar work as used in my near-infrared (NIR) drone research. </span></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh_Qzefmj7V3eS8Y5_8FsnvuUDaWyZ2Z3zKNEj1KWj1J5OUIY4OAm4nW2Z5n1nsObxlGkP8HH31NT4APjk-mUraMfAmPsX43dPa9tVU0ft71_gtlm0A9M1m_BsKX-FcWTGVTpOElpsmie9g/s2048/InfraBlueNDVI4.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1330" data-original-width="2048" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh_Qzefmj7V3eS8Y5_8FsnvuUDaWyZ2Z3zKNEj1KWj1J5OUIY4OAm4nW2Z5n1nsObxlGkP8HH31NT4APjk-mUraMfAmPsX43dPa9tVU0ft71_gtlm0A9M1m_BsKX-FcWTGVTpOElpsmie9g/s320/InfraBlueNDVI4.jpg" width="320" /></a></div><div class="separator" style="clear: both; text-align: center;">Ultraviolet Reflectance with an NDVI-style Key</div><div><br /></div>All coding available through my GitHub Repository - https://github.com/MuonRay/Ultraviolet_Image_Python_Processing_Codes<br /><div dir="ltr" style="text-align: left;" trbidi="on"><br /></div><div dir="ltr" style="text-align: left;" trbidi="on"><br /></div>
MuonRayhttp://www.blogger.com/profile/03712859045968965104noreply@blogger.com0tag:blogger.com,1999:blog-5996268492156364201.post-46620808403734084282020-09-07T10:13:00.001-07:002020-09-07T10:13:42.205-07:00Ion Propulsion of Magnetic Levitating Graphene E-Sail: "Tesla-Kinesis"<p></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiwQIiT9GL1HfvQ08pgTjFF0qQNx1i8Rzz3wjeOhohyphenhyphenRoLtxm5zt4D_iOoSpIwUJuqqzyHmnHZ68VI9KW0OdaLAhJsLKq5D_QDkgDOfVEhsChTqYabFY87-IoxaSdkA7UpBOB4vBjzXI74O/s1616/157393863940680297+%25281%2529+-+Copy.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1140" data-original-width="1616" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiwQIiT9GL1HfvQ08pgTjFF0qQNx1i8Rzz3wjeOhohyphenhyphenRoLtxm5zt4D_iOoSpIwUJuqqzyHmnHZ68VI9KW0OdaLAhJsLKq5D_QDkgDOfVEhsChTqYabFY87-IoxaSdkA7UpBOB4vBjzXI74O/s320/157393863940680297+%25281%2529+-+Copy.png" width="320" /></a></div><br /><span style="background-color: #f9f9f9; color: #030303; font-family: Roboto, Arial, sans-serif; font-size: 14px; white-space: pre-wrap;"><br /></span><p></p><p><span style="background-color: #f9f9f9; color: #030303; font-family: Roboto, Arial, sans-serif; font-size: 14px; white-space: pre-wrap;">In this short experimental demonstration, I share a concept of induced ion propulsion using positive ions that create a wind that can push a thin-film of graphene that is kept levitating at effectively zero-G using a rare-earth magnetic track. This is, in effect, an ion "E-sail" (electric sail) that can capture the momentum of the ions emitted and transfer them into motion in the direction of the ion flow. </span></p><p><span style="background-color: #f9f9f9; color: #030303; font-family: Roboto, Arial, sans-serif; font-size: 14px; white-space: pre-wrap;"></span></p><div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen="" class="BLOG_video_class" height="266" src="https://www.youtube.com/embed/nA5r6al0Bjw" width="320" youtube-src-id="nA5r6al0Bjw"></iframe></div><br />
The ion source is a modified Tesla coil, a high voltage source, that causes breakdown of air above a sharp steel tip creating a streak of positive ions that move away from the source in a direction toward the sample being probed and propelled. <p></p><p><span style="background-color: #f9f9f9; color: #030303; font-family: Roboto, Arial, sans-serif; font-size: 14px; white-space: pre-wrap;"></span></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiEnYHY8yCq4DrqCOohS-P0nQxA5HdiV-7HrqpH0-Jg8MxQ5S2aOZX3st2YMTVQNrcbZcEecg4DY8qjT9WnNqZJCX9iz_D0K76ooceH2UV-w9dGhhKThp3Fnvh1uuc6gaOzYG-9xjQXqKqh/s948/Tesla+Ion+Propulsion+of+Graphene.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="487" data-original-width="948" height="256" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiEnYHY8yCq4DrqCOohS-P0nQxA5HdiV-7HrqpH0-Jg8MxQ5S2aOZX3st2YMTVQNrcbZcEecg4DY8qjT9WnNqZJCX9iz_D0K76ooceH2UV-w9dGhhKThp3Fnvh1uuc6gaOzYG-9xjQXqKqh/w500-h256/Tesla+Ion+Propulsion+of+Graphene.jpg" width="500" /></a></div><br />
This system is a simulation of the solar-wind that is whipped up by the high temperatures and magnetic activity of the stars themselves. The Sun produces a significant solar wind, made up of protons and helium nuclei, which are emitted at high velocities from the solar atmosphere, the corona, and solar surface during solar flares and coronal mass ejections. The energy that is emitted in these eruptions, translated into the stream of high velocity charged ions, is perhaps the greatest free source of space propulsion and could, potentially, carry spacecraft equipped with massive solar sails to the outer reaches of the solar system at speeds that would be impossible to achieve using chemical propulsion or gravity assists and without the need for onboard propellant. <p></p><p><span style="background-color: #f9f9f9; color: #030303; font-family: Roboto, Arial, sans-serif; font-size: 14px; white-space: pre-wrap;"></span></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgNyP9iHG9p8Rgl7f7mdMIi0OlQy1fbZIergNdcqYYXncECtl-7SNK3BHC6vw57QTukCh2t-A9VQamSjPnhZfsOpqSgVqxNgu-fW_ghaY5eVign43ifMvN6KfWYQgL1iXAtxu7QJKf2-_7d/s1408/GrapheneESailByProtonCapture.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="628" data-original-width="1408" height="224" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgNyP9iHG9p8Rgl7f7mdMIi0OlQy1fbZIergNdcqYYXncECtl-7SNK3BHC6vw57QTukCh2t-A9VQamSjPnhZfsOpqSgVqxNgu-fW_ghaY5eVign43ifMvN6KfWYQgL1iXAtxu7QJKf2-_7d/w500-h224/GrapheneESailByProtonCapture.png" width="500" /></a></div><br />
Indeed, interstellar space travel could be achieved using this effect over the more energy intensive photon-assisted propulsion which is also another interesting avenue of research which could utilize the high durability and strength of graphene material. Graphene also has the advantage of being highly resistant to radiation and the electrical current induced by ion capture may itself be used as an energy source for the spacecraft with a large enough sail.
In any case the effect itself is interesting and demonstrates, if nothing else, the principle of converting electrical energy into kinetic energy and the visual demonstrations of the concept of an ion wind.<p></p><p><br /></p>MuonRayhttp://www.blogger.com/profile/03712859045968965104noreply@blogger.com0tag:blogger.com,1999:blog-5996268492156364201.post-60700623000710833172020-08-26T07:25:00.003-07:002020-08-26T08:06:27.568-07:00Hybrid Wearable Energy Harvester -Thermal and Solar Energy Harvesting<iframe allowfullscreen="" frameborder="0" height="270" src="https://www.youtube.com/embed/stGh3__ih6c" width="480"></iframe><br /><br />
<br /><br />
Thank you for your support for this and other projects on my channel: https://www.patreon.com/muonray<br /><br />
I have been working on this flexible wearable hybrid energy harvester prototype for some time as a follow up to a previous version that used a new flexible thermoelectric material that allows one to convert body-heat to electricity in a wearable design, as shown here: <div><br /></div><div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen="" class="BLOG_video_class" height="266" src="https://www.youtube.com/embed/TtVinWOS1DU" width="320" youtube-src-id="TtVinWOS1DU"></iframe></div><br /><div><br /><br />
Here I present to you a new technology showcase with a lot of potential over the next few years - wearable and flexible hybrid energy harvesters that can acquire multiple energy sources simultaneously and convert them into useful electrical energy for charging and lighting applications. The possibilities are large with this kind of electronics and I have no doubt that companies will begin to develop wearable appliances similar to this over the next few years especially as new generations of energy efficient smartphones, watches and even clothing integrated electronics become more widespread.<br /><br />The South-Korean Company TegWay have developed flexible Peltier-effect thermoelectric heater/cooler (TEC) modules for use in augmented and immersive virtual reality applications - giving game controllers the ability to create feelings of cold or heat for immersive movie or gaming experiences for example. The same technology can also be used to create TEG modules for use in energy harvesting applications.<br /><br />
This kind of flexible energy harvester kind of a silver bullet for energy harvesting from body heat and also from waste heat from pipes and circular objects such as pipes which would have been difficult to attach a monolithic rigid TEG module and efficiently power devices from, such as IOT sensors and such, and to conserve energy loss in general from a system. In effect there is never such a thing as "free" energy, the waste heat being converted to electricity has its origins in either metabolic processes from a human being in wearable energy harvesting or as waste heat from a machine or power source.<br /><br /><br />
It may be possible to see flexible thermoelectric generator modules stitched into clothing in the near future for powering smart devices such as phones and watches. More interesting applications still involve the development of thermoelectric energy generator (TEG) suits for expeditions in remote places to power electronics for geotagging or monitoring sensors. <div><br /></div><div>Perhaps such TEG Suits could be put to use out in the blustery conditions at sea, in mountains and the cold polar regions of the Earth or even developed into spacesuits for generating on-demand electricity for astronauts exploring Mars!</div><div><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhvYi6rvWI0v0EamUkHwyRjmyHx7ce2AG-1i41KppHadLi5tBXQzH4Jd66h8Of4Onm4wzuL0XB5jDg7wZ9NzGLL_fPhLtZa8y6C9WZCbedcLlsXh-HlbQcBXQos-u4uFILDXzzj-34MXKp1/s1190/e66efc9fad467e8d34996c003045c26d.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="662" data-original-width="1190" height="285" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhvYi6rvWI0v0EamUkHwyRjmyHx7ce2AG-1i41KppHadLi5tBXQzH4Jd66h8Of4Onm4wzuL0XB5jDg7wZ9NzGLL_fPhLtZa8y6C9WZCbedcLlsXh-HlbQcBXQos-u4uFILDXzzj-34MXKp1/w512-h285/e66efc9fad467e8d34996c003045c26d.jpg" width="512" /></a></div><br /><div><br /></div><div>One can think of an suit that harvests Energy in cold environments being a huge advantage in places such as Antarctica or in the frigid conditions of the planet Mars where the temperature differences between the human body inside the suit and the subzero temperature outside the suit could be used to create a significant amount of power for explorers.</div><div><br />
Combining the technology with an efficient flexible thin-film solar panel adds for more energy harvesting capacity and does not significantly affect the heat exchange mechanism as long as the solar cell used is of an extremely thin film. Positioned with the back of the element is Coated with Silver Thermal Paste and combined with Flexible Silicone allows for even more efficient Heat Exchange, creating a temperature differential across the TEG element and allowing the solar energy to be harvested while the element is in contact with a warm body or other heat source, for example electronic equipment that gets warm with use. <br /><br /><br />
Another application could be in the geotagging of warm-blooded animals using a system that does not depend on batteries recharging from solar panels and instead uses the animals own body heat to power the device</div><div><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjA8eC6sb5U3d3u8vuwOqZ8K2Qd5mFuJJmpLh4b9Ab5X28ZkBQBigla1TFcMMRRD_jhYStQhh8BDVKDqutTkB-jhTVKfH5aoKiwuk48RIwUrhmTMzJypwYMWC5hJk36Yj-z9s9xuzk6mALj/s480/solar+animal+tag.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="360" data-original-width="480" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjA8eC6sb5U3d3u8vuwOqZ8K2Qd5mFuJJmpLh4b9Ab5X28ZkBQBigla1TFcMMRRD_jhYStQhh8BDVKDqutTkB-jhTVKfH5aoKiwuk48RIwUrhmTMzJypwYMWC5hJk36Yj-z9s9xuzk6mALj/s0/solar+animal+tag.jpg" /></a></div><br /><div><br /></div><div><br /></div><div>Perhaps such hybrid energy harvested-powder gps tags could find use for maritime applications or in colder regions of the earth, for more see this article here:</div><div><a href="https://phys.org/news/2020-07-solar-powered-animal-tracker-animals-wild.html">https://phys.org/news/2020-07-solar-powered-animal-tracker-animals-wild.html</a><br /><br /><br />
For the moment research is ongoing in developing applications and this still remains an interesting demonstration to observe the ability to transform one type of energy, thermal energy, into another, electrical energy.<br /><br />
<br /></div></div>MuonRayhttp://www.blogger.com/profile/03712859045968965104noreply@blogger.com0tag:blogger.com,1999:blog-5996268492156364201.post-65763024481963307692020-08-24T15:34:00.003-07:002020-08-24T15:35:17.963-07:00MuonRay Enterprises Now Based in The Canary Islands For New Research Opportunities! <p></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi9Q3X8Fn8rOhwlF9lAuwnFUgFXU6e0lrnBMT4YyR5oaXwBHuRaDGAW-VKMqpFWfKNJpMQM84IyKXcqoKrRi4MpxUA0MSFlPlHji1EXbhfztwwn3-nY8JwO0u6oupvILFfq-KHsfqMpJr81/s476/ezgif-4-cd0ab4c5eaef.gif" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="476" data-original-width="476" height="305" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi9Q3X8Fn8rOhwlF9lAuwnFUgFXU6e0lrnBMT4YyR5oaXwBHuRaDGAW-VKMqpFWfKNJpMQM84IyKXcqoKrRi4MpxUA0MSFlPlHji1EXbhfztwwn3-nY8JwO0u6oupvILFfq-KHsfqMpJr81/w305-h305/ezgif-4-cd0ab4c5eaef.gif" width="305" /></a></div> <p></p><p>I am pleased to announce that starting this week I am now based in Las Palmas in Gran Canaria for an exciting new series of research projects involving the testing of new imaging technologies for use in drones and astronomical optics.</p><p>Some of the new projects I am involving myself in include further testing of infrared optics for use in drone cameras and a brand new technique I have been developing that uses Ultraviolet filters to allow for real-time Near-Ultraviolet Drone Filming and Photography.</p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiW0WXjJxUPC2jCKNrmN91Ll5txzrk0cH18yOam_nTzi4cqFmioQ6v3aQcy6QMKwfI3xQfO_e8WYsa2Wj4QNYwGm4vN_Do7HFTUQnJKg-sqQriBBv0MtqSRqo9Js09qcUXU80UVUIs00-yr/s5472/DJI_0829.JPG" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="3078" data-original-width="5472" height="288" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiW0WXjJxUPC2jCKNrmN91Ll5txzrk0cH18yOam_nTzi4cqFmioQ6v3aQcy6QMKwfI3xQfO_e8WYsa2Wj4QNYwGm4vN_Do7HFTUQnJKg-sqQriBBv0MtqSRqo9Js09qcUXU80UVUIs00-yr/w512-h288/DJI_0829.JPG" width="512" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><i>A Near-UV test photo - The Dry Conditions and high UV radiance of the Canary Islands should allow for excellent filming conditions. </i></div><p>I am confident that this new technique of imaging will open up a whole new field of environmental imaging in the near-UV that will be an exclusively drone-based imaging technique for remote sensing.</p><p><br /></p><p>I am also working on new infrared imaging techniques for use in astronomy using cryocooled sensors I have been developing in Ireland for use in astrophysical imaging and filming in the Long-Wavelength Infrared (LWIR) that I have been developing in Ireland for some years now. A video of such a cryocooler I will be using with a LWIR CCD sensor is shown below:</p><div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen="" class="BLOG_video_class" height="266" src="https://www.youtube.com/embed/aaEYD6sw408" width="320" youtube-src-id="aaEYD6sw408"></iframe></div><p><br /></p><p>Thank you very much for all the supporters of these projects in the past and the future is looking very bright indeed for research for the next couple of months and hopefully beyond here in Gran Canaria!</p><p><br /></p>MuonRayhttp://www.blogger.com/profile/03712859045968965104noreply@blogger.com0