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

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.

Get Rid of Nonlocality from Quantum Physics

“…This paper should not be considered as directed against attempts to go beyond QM, by introducing “hidden variables”. However, in such attempts, one has to take into account the basic principles of QM an especially the complementarity principle (see the recent article of Khrennikov and Alodjants [ 18 ]). One also has to take into account the lessons of 19th century physics in the period of transition from Newtonian mechanics to field theory ( Section 6.4 ).…”

“…As emphasized above, here I proceed by using solely the formalism of QM, cf. with probabilistic analysis of the incompatibility interpretation of the Bell type inequalities in [ 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 ] and especially [ 17 ]— the probabilistic version of the present paper. See also the recent preprint of Griffiths [ 25 ], where incompatibility of quantum observables is emphasized; see the recent works of Boughn [ 26 , 27 ], where the nonlocality viewpoint on quantum theory is critically analyzed and the role of the ontological vs. information interpretations of the wave function in discussions on “quantum nonlocality” is emphasized.…”

Probability Representation of Quantum Mechanics Where System States Are Identified with Probability Distributions

“…The statistical properties of qubit quantum observables were discussed in relation with the classical statistical properties of the classical-like dichotomic random variables. The different aspects of the relation of the probability theory with properties of quantum or quantum-like states were discussed in [45][46][47][48][49]. We will consider other examples of the connection of the path integral with star-product schemes and entropic inequalities for quantum systems based on the probability representation of quantum mechanics in future publications.…”

A Generic Model for Quantum Measurements

“…As explained in the articles quoted above, this dichotomy has the great virtue to eliminate the “wave-packet collapse” as it appears in the usual approach (because there is no wave-packet any more! ), as well as any “instantaneous influence at a distance” in Bell test experiments [ 15 , 16 , 17 , 18 ]. Nevertheless, a remaining open issue is the physical “cut”, i.e., the physical separation between the system and the context.…”