Bohm-Bell type experiments: Classical probability approach to (no-)signaling and applications to quantum physics and psychology

Andrei Khrennikov; Alexander Alodjants

doi:10.3390/e21020157

paper

Bohm-Bell type experiments: Classical probability approach to (no-)signaling and applications to quantum physics and psychology

arXiv:1812.10826 · doi:10.3390/e21020157

Abstract

We consider the problem of representation of quantum states and observables in the framework of classical probability theory (Kolmogorov's measure-theoretic axiomatics, 1933). Our aim is to show that, in spite of the common opinion, correlations of observables $A_{1}, A_{2}$ and $B_{1}, B_{2}$ involved in the experiments of the Bohm-Bell type can be expressed as correlations of classical random variables $a_{1}, a_{2}$ and $b_{1}, b_{2} .$ The crucial point is that correlations $⟨ A_{i}, B_{j} ⟩$ should be treated as conditional on the selection of the pairs $(i, j) .$ The setting selection procedure is based on two random generators $R_{A}$ and $R_{B} .$ They are also considered as observables, supplementary to the "basic observables" $A_{1}, A_{2}$ and $B_{1}, B_{2} .$ These observables are absent in the standard description, e.g., in the scheme for derivation of the CHSH-inequality. We represent them by classical random variables $r_{a}$ and $r_{b} .$ Following the recent works of Dzhafarov and collaborators, we apply our conditional correlation approach to characterize (no-)signaling in the classical probabilistic framework. Consideration the Bohm-Bell experimental scheme in the presence of signaling is important for applications outside quantum mechanics, e.g., in psychology and social science.

Bohm-Bell type experiments: Classical probability approach to (no-)signaling and applications to quantum physics and psychology

Abstract

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