Tuesday, 5 January 2016

"No Cloning" and "Entanglement Monogamy" - Quantum Theory to Quantum Heuristics


The "no cloning" theorem, is an axiom in quantum mechanics which states that it is impossible for a single quantum variable, such as spin of an electron or polarization of an photon, to be maximally entangled with 2 other counter variables, i.e. two other spins or polarization states, simultaneously.

This can be derived by theory, but experimentally leads to something called "the monogamy of entanglement". Since we are dealing with quantum particles, the link between theory and experiment naturally deals with representations in information theory, in other words to carry out any experiments at all we need to develop algorithms and heuristic strategies to prove or disprove the theorem in the laboratory.

Let us first set up the framework for this test. The 2 quantum variable case is of course the basis of our understanding of entanglement and we have covered this in a previous article with an interesting example. However, this time we will be using a much more diagrammatic representation.

In each case, what we are doing is in effect a test of the CHSH Inequality. In quantum physics, the CHSH inequality can be used as our heuristic strategy in the proof of Bell's theorem, which states that certain consequences of entanglement in quantum mechanics cannot be reproduced by local hidden variable theories. In our cases, this means that the local hidden variables involved in shared random information exchanged between 2 partners, Alice and Bob, will not have the same maximum win probability (3/4, i.e. 75%) as that with shared quantum entangled information (which is much greater at 0.8536 = 85.36%) - we will prove this using CHSH in our example with a 2-state system, which itself proves Bell's theorem, and then in the same vein use our "Monogamy of Entanglement" with a 3-state system to prove the "no-cloning" theorem.

CHSH stands for John Clauser, Michael Horne, Abner Shimony, and Richard Holt, who described it in a much-cited paper published in 1969 (Clauser et al., 1969) They derived the CHSH inequality, which, as with John Bell's original inequality (Bell, 1964), to act as a constraint on the statistics of "coincidences" in a Bell test experiment which is necessarily true if there exist underlying local hidden variables (local realism) which would be the case given the fact that in dealing with single quantum emissions from sources such as lasers or atoms there will always be effects to decohere the source.

The Setup:

So to set things up lets consider the 2-state quantum entangled system of polarized photons:

Photon polarization states:

Measurement: Questions of the form,
Are you

Heisenberg Uncertainty principle: No photon will yield
a definite answer to
both measurements:

General Rule:

Quantum state: α      
Measurement: An orthonormal basis {v1, …, vN}

Pr[v|α] = |hvi,αi|2  


1. Alice chooses a random
sequence of bits and encodes
each one using either:

2. Bob randomly chooses
to measure with either:

3.They publically reveal their
choice of axes and discard
pairs that don’t match.

4. If remaining bits are
perfectly correlated, then
they are also secret.

Defining the Entanglement:

which is equivalent to:

i.e. Alice and Bob share state: 

Using the General rule: 
Pr[A,B observe v,w | state α] = |hv­w, αi|cos2(θ) / 2

Alice measures {v1,v2}, Bob measures {w1,w2}.

Pr[w1|v1] = cos2(θ)    Pr[v1] = 1/2
Pr[w1|v2] = sin2(θ)    Pr[v2] = 1/2

Total Mapped Outcomes:

Alice measures:

Bob Measures:

Winning pairs are at angle π/8

Losing pairs are at angle 3π/8  

Pr[win] = cos2(π/8) ≈ 0.854 

As stated, a more mathematical (and less diagrammatic) example of this is shown here with respect to prisoner's dilemma:

Monogamy of Entanglement:

Now we can use the exact same setup and formulae to represent a 3-state quantum entangled system to demonstreate the "Monogamy of Entanglement" which is the way to prove the "No-Cloning" Theorem in the same way as the CHSH test is a way to prove the Bell's Theorem. In the 3 quantum variable case, the heuristic link between cloning and entanglement monogamy will continue to be our entangled photons.

One then follows a quantum teleportation protocol with an unknown state, Charlie, targeting polarization state copies of the unknown state hoping they will appear on the other two polarization states, with the quality of the copies depending on how much entanglement was in the original state, the more entanglement shared between say Alice and Bob, the better the quality of the copy of the unknown state Charlie holds.

The General rule is then modified to: 

max Pr[AB] + Pr[AC] =
max Pr[x
y = ab] + Pr[xz = ac]
< 2cos2(π/8)

The question to ask then is, given a state of A, B, ,represented as:

What is the state of AB?  or AC?

To answer this we must measure  C.

Outcomes {0,1} have probability:

AB are left with:

This leads to the General Entanglement Monogamy Relation:

The distributions over AB and AC cannot both be entangled.

This leads to the conclusion that if a particular quality of cloning is impossible (in particular if the unknown state cannot appear perfectly at both polarization states A and B), a certain degree of entanglement must be impossible (no state can have maximal entanglement between A and B as well as between spins A and C).

(with the same basic criterion if we consider complimentary spin states of particles such as electrons or atoms)

Hence, this protocol can be implemented on a quantum circuit, thus verifying the no-cloning principle experimentally in the lab with entangled bosons, such as photons, or fermions, such as electrons and certain fermionic atom condensates, as well as bosonic atom condensates which essentially covers all the types of particles and fields. At the quantum level, cloning is impossible, both in theory and experiment, and can therefore only be circumvented*.

* Now this is where it gets very interesting, as such circumventions to the "monogamy of entanglement" principle actually must occur near the event horizon of a black hole - and by extension must have happened at the Big Bang singularity itself - this has led to a paradox in physics, known as the "Quantum Firewall Paradox" or "Black Hole Information Paradox" or "Black Hole Firewall Paradox" (interesting and strange sounding names!) which you can read more about here:
Black Hole Information Firewall Paradox Explained