Does CHSH Inequality Test the Model of Local Hidden Variables

It is pointed out that the local hidden variables model of Bell and Clauser-Horne-Shimony-Holt (CHSH) gives $||\leq 2\sqrt{2}$ or $||\leq 2$ 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 $||\leq 2$ does not provide a reliable test of the $d=4$ local non-contextual hidden variables model. To achieve $||\leq 2$ 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 $||\leq 2$, but no more a proper hidden variables model in $d=4$. The conventional CHSH inequality $||\leq 2$ 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.