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arXiv:1808.02379 · Journal of Mathematical Psychology

Classical versus quantum probability: Comments on the paper “On universality of classical probability with contextually labeled random variables” by E. Dzhafarov and M. Kon

Andrei Yu. Khrennikov

Published Jan 29, 2019• 9 citations
arXivHTMLPDF
DOI
10.1016/j.jmp.2018.10.005

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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.
Several principled measures of contextuality have been proposed for general systems of random variables (i.e. inconsistently connected systems). One such measure is based on quasi-couplings using negative probabilities (here denoted by CNT 3 , Dzhafarov & Kujala, 2016 Quantum interaction ). Dzhafarov & Kujala (Dzhafarov & Kujala 2019 Phil. Trans. R. Soc. A 377 , 20190149. ( doi:10.1098/rsta.2019.0149 )) introduced a measure of contextuality, CNT 2 , that naturally generalizes to a measure of non-contextuality. Dzhafarov & Kujala (Dzhafarov & Kujala 2019 Phil. Trans. R. Soc. A 377 , 20190149. ( doi:10.1098/rsta.2019.0149 )) additionally conjectured that in the class of cyclic systems these two measures are proportional. Here we prove that conjecture is correct. Recently, Cervantes (Cervantes 2023 J. Math. Psychol . 112 , 102726. ( doi:10.1016/j.jmp.2022.102726 )) showed the proportionality of CNT 2 and the Contextual Fraction measure introduced by Abramsky & Brandenburger (Abramsky & Brandenburger 2011 New J. Phys . 13 , 113036. ( doi:10.1088/1367-2630/13/11/113036 )). The present proof completes the description of the interrelations of all contextuality measures proposed within or translated into the Contextuality-by-Default framework so far as they pertain to cyclic systems. This article is part of the theme issue ‘Quantum contextuality, causality and freedom of choice’.