Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 66 / 3

The EPR criterion of reality proposes that a quantity has an element of physical reality when its value can be predicted with certainty without disturbing the system. It is a sufficient criterion, not a claim that every uncertain prediction is unreal. In the original locality argument, measuring one member of a separated correlated pair allows such a prediction for the other, whose physical condition is assumed unaffected by the remote choice.

Combining the EPR criterion of reality with perfect predictive correlations and locality motivates pre-existing values for the locally selectable observables. The general local hidden-variable theory additionally assumes a complete shared variable $\lambda$ and conditional factorizationThe second condition is measurement independence. The reality criterion alone does not prove this factorization for arbitrary imperfect correlations: it motivates the local completion whose consequences are being tested.

A deterministic local hidden-variable model may be used without loss of generality by putting local random seeds into $\lambda$. For each $\lambda$, write its setting responses as $A,A^{\prime },B,B^{\prime }\in \{-1,1\}$. Integrating with the same setting-independent distribution and using the triangle inequality givesExactly one of $B-B^{\prime }$ and $B+B^{\prime }$ vanishes and the other has magnitude two. Hence the requested CHSH bound is two. No assumption about the spacelike quantum state enters this local-model derivation.

The equal mixture of $|{\Phi }^{+}⟩$ and $|{\Phi }^{-}⟩$ cancels their off-diagonal coherence, leaving the separable quantum stateA trivial LHV model chooses a shared random bit $j$, uniformly zero or one, and prepares both local qubits in $|j⟩$. Each party uses the local Born rule for its own chosen spin direction, with independent local random seeds if a deterministic description is desired. This reproduces the entire joint measurement distribution, not merely the $z$-axis correlation.

Since the state has an explicit local hidden-variable theory for these local measurements, it cannot violate the CHSH inequality, by the derivation in part (b). Classical correlation of the shared preparation bit is sufficient; the cancellation of coherence removes the entanglement.

This is the dephased Bell-state mixture with coherence parameter $2x$. The Pauli correlation tensor isso spin measurements along unit vectors have $E(\mathbf{a},\mathbf{b})={\mathbf{a}}^{T}T\mathbf{b}$. ChooseThen $E(a,b)-E(a,b^{\prime })=2/\sqrt{1+4x^2}$ and $E(a^{\prime },b)+E(a^{\prime },b^{\prime })=8x^2/\sqrt{1+4x^2}$. The CHSH inequality expression is thereforeThese axes also attain the CHSH optimum from the correlation tensor, since the two largest eigenvalues of $T^{T}T$ are $1$ and $4x^2$. At $x=1/2$ the value is $2\sqrt{2}$; at $x=0$ the explicit LHV model in part (c) applies. Thus exactly $0<x\le 1/2$ has correlations incompatible with a local hidden-variable description.

Arbitrarily small positive coherence suffices in this particular one-parameter family, with optimally chosen axes; the violation above two tends to zero quadratically as $x\to 0$. This does not imply that every entangled mixed state violates a CHSH inequality.