Wigner's inequality for conditional probabilities and nonexistence of a realistic model for the two dimensional Hilbert space

TL;DR

This paper analyzes Bell's probabilistic assumptions, demonstrating that classical probability theory cannot account for quantum correlations in two-dimensional systems, and introduces a Wigner-type inequality for conditional probabilities.

Contribution

It reveals that Bell's model does not align with classical probability axioms and derives a new inequality showing the incompatibility of classical and quantum probabilities in simple systems.

Findings

Bell's model does not use classical probability axioms

A Wigner-type inequality for conditional probabilities is established

Quantum and classical predictions differ in two-dimensional systems

Abstract

We perform the analysis of probabilistic assumptions of Bell's approach. We emphasize that J. Bell wrote about probability without to specify the concrete axiomatics of probability theory. The careful analysis demonstrated that (surprisingly) J. Bell did not apply the classical probability model (Kolmogorov) to describe ``classical physical framework.'' In fact, he created his own probabilistic model and compared it with the quantum one. The crucial point is that J. Bell did not pay attention to {\it conditional probabilities.} We show that conditional probability in his model cannot be defined by classical Bayes' formula. We also use the approach based on Bell-type inequalities in the conventional probabilistic approach, Kolmogorov model. We prove an analog of Wigner's inequality for conditional probabilities and by using this inequality show that predictions of the conventional and quantum probability models disagree already in the case of noncomposite systems (even in the two dimensional case).