CHSH inequality: Quantum probabilities as classical conditional probabilities

Abstract

The celebrating theorem of A. Fine implies that the CHSH inequality is violated if and only if the joint probability distribution for the quadruples of observables involved the EPR-Bohm-Bell experiment does not exist, i.e., it is impossible to use the classical probabilistic model (Kolmogorov, 1933). In this note we demonstrate that, in spite of Fine's theorem, the results of observations in the EPR-Bohm-Bell experiment can be described in the classical probabilistic framework. However, the "quantum probabilities" have to be interpreted as conditional probabilities, where conditioning is with respect to fixed experimental settings. Our approach is based on the complete account of randomness involved in the experiment. The crucial point is that randomness of selections of experimental settings has to be taken into account. This approach can be applied to any complex experiment in which statistical data are collected for various (in general incompatible) experimental settings. Finally, we emphasize that our construction of the classical probability space for the EPR-Bohm-Bell experiment cannot be used to support the hidden variable approach to the quantum phenomena. The classical random parameter $\omega$ involved in our considerations cannot be identified with the hidden variable $\lambda$ which is used the Bell-type considerations.