Does CHSH inequality test the model of local hidden variables?

TL;DR

The paper argues that the traditional CHSH inequality does not reliably test local hidden variables models in a four-dimensional quantum system due to non-linearity issues, and it clarifies the inequality's true scope in quantum state characterization.

Contribution

It demonstrates that the conventional CHSH inequality only characterizes separable states and cannot serve as a definitive test for local hidden variables in four-dimensional systems.

Findings

The CHSH inequality does not test local hidden variables in d=4 systems.

Linearity requirement converts local models to factored non-contextual models.

CHSH inequality characterizes separable states, not hidden variables in d=4.

Abstract

It is pointed out that the local hidden variables model of Bell and Clauser-Horne-Shimony-Holt (CHSH) gives $|<B>|\leq 2\sqrt{2}$ or $|<B>|\leq 2$ for the quantum CHSH operator $B={\bf a}\cdot {\bf \sigma}\otimes ({\bf b}+{\bf b}^{\prime})\cdot {\bf \sigma} +{\bf a}^{\prime}\cdot{\bf \sigma}\otimes ({\bf b}-{\bf b}^{\prime})\cdot{\bf \sigma} $ depending on two different ways of evaluation, when it is applied to a $d=4$ system of two spin-1/2 particles. This is due to the failure of linearity, and it shows that the conventional CHSH inequality $|<B>|\leq 2$ does not provide a reliable test of the $d=4$ local non-contextual hidden variables model. To achieve $|<B>|\leq 2$ uniquely, one needs to impose a linearity requirement on the hidden variables model, which in turn adds a von Neumann-type stricture. It is then shown that the local model is converted to a factored product of two non-contextual $d=2$ hidden variables models. This factored product implies pure separable quantum states and satisfies $|<B>|\leq 2$, but no more a proper hidden variables model in $d=4$. The conventional CHSH inequality $|<B>|\leq 2$ thus characterizes the pure separable quantum mechanical states but does not test the model of local hidden variables in $d=4$, to be consistent with Gleason's theorem which excludes non-contextual models in $d=4$. This observation is also consistent with an application of the CHSH inequality to quantum cryptography by Ekert, which is based on mixed separable states without referring to hidden variables.