Mathematical and physical meaning of the Bell inequalities

TL;DR

This paper explores the mathematical and physical significance of Bell inequalities, linking them to triangle inequalities among random variables and discussing their implications for quantum nonlocality and contextuality.

Contribution

It provides a novel interpretation of Bell inequalities through distance functions and clarifies their role in quantum mechanics and local realism.

Findings

Bell inequalities relate to triangle inequalities among random variables.

They serve as necessary conditions for noncontextual hidden variables models.

The paper discusses Bell inequalities' connection to quantum entanglement and contextuality.

Abstract

It is shown that the Bell inequalities are closely related to the triangle inequalities involving distance functions amongst pairs of random variables with values $\left\{ 0,1\right\} $. A hidden variables model may be defined as a mapping between a set of quantum projection operators and a set of random variables. The model is noncontextual if there is a joint probability distribution. The Bell inequalities are necessary conditions for its existence. The inequalities are most relevant when measurements are performed at space-like separation, thus showing a conflict between quantum mechanics and local realism (Bell's theorem). The relations of the Bell inequalities with contextuality, Kochen-Specker theorem, and quantum entanglement are briefly discussed.