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 $|<B>|\leq 2\sqrt{2}$ 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. 
Kazuo Fujikawa
quant-ph hep-th math-ph math.MP