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Entropy 2019, 21(2), 157; https://doi.org/10.3390/e21020157

Classical (Local and Contextual) Probability Model for Bohm–Bell Type Experiments: No-Signaling as Independence of Random Variables

1
Mechanics and Optics (ITMO) Department, National Research University for Information Technology, St. Petersburg 197101, Russia
2
International Center for Mathematical Modeling in Physics and Cognitive Sciences, Linnaeus University, Växjö, SE-351 95, Sweden
*
Author to whom correspondence should be addressed.
Received: 10 December 2018 / Revised: 29 January 2019 / Accepted: 4 February 2019 / Published: 8 February 2019
(This article belongs to the Special Issue Towards Ultimate Quantum Theory (UQT))
PDF [312 KB, uploaded 8 February 2019]

Abstract

We start with a review on classical probability representations of quantum states and observables. We show that the correlations of the observables involved in the Bohm–Bell type experiments can be expressed as correlations of classical random variables. The main part of the paper is devoted to the conditional probability model with conditioning on the selection of the pairs of experimental settings. From the viewpoint of quantum foundations, this is a local contextual hidden-variables model. Following the recent works of Dzhafarov and collaborators, we apply our conditional probability approach to characterize (no-)signaling. Consideration of the Bohm–Bell experimental scheme in the presence of signaling is important for applications outside quantum mechanics, e.g., in psychology and social science. The main message of this paper (rooted to Ballentine) is that quantum probabilities and more generally probabilities related to the Bohm–Bell type experiments (not only in physics, but also in psychology, sociology, game theory, economics, and finances) can be classically represented as conditional probabilities.
Keywords: quantum versus classical probability; Bohm–Bell type experiments in physics and psychology; localty; contextual hidden-variables models; (no-)signaling; random generators; selection of experimental settings, conditional probability quantum versus classical probability; Bohm–Bell type experiments in physics and psychology; localty; contextual hidden-variables models; (no-)signaling; random generators; selection of experimental settings, conditional probability
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Khrennikov, A.; Alodjants, A. Classical (Local and Contextual) Probability Model for Bohm–Bell Type Experiments: No-Signaling as Independence of Random Variables. Entropy 2019, 21, 157.

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