The correlation Bell inequalities

We consider the Bell and Bell–Clauser–Horne–Shimony–Holt inequalities for two-particle spin states. It is known that these inequalities are violated in experimental verification. We show that this can be explained because these inequalities are proved for correlation functions of random variables that are totally unrelated to one another, while the verification is done using correlation functions in which random variables refer to a pair of particles forming a two-particle state. In the case of entangled states, these random functions are dependent, and their correlation coefficient is nonzero. We give inequalities that explicitly involve this correlation coefficient. For factorable and separable states, these inequalities coincide with the standard Bell and Bell–Clauser–Horne–Shimony–Holt inequalities.