Extended Boole-Bell inequalities applicable to quantum theory

TL;DR

This paper revises Bell's inequalities by considering extended Boole-Bell inequalities, demonstrating their validity in both classical and quantum models, and explaining apparent violations through incomplete premises.

Contribution

It introduces extended Boole-Bell inequalities that unify classical and quantum models, clarifying the origin of apparent contradictions in quantum theory.

Findings

Both EPR-Bohm and Leggett-Garg experiments obey the extended inequalities.

Apparent violations are due to incomplete premises in derivations.

The inequalities are applicable to macroscopic quantum coherence.

Abstract

We address the basic meaning of apparent contradictions of quantum theory and probability frameworks as expressed by Bell's inequalities. We show that these contradictions have their origin in the incomplete considerations of the premisses of the derivation of the inequalities. A careful consideration of past work, including that of Boole and Vorob'ev, has lead us to the formulation of extended Boole-Bell inequalities that are binding for both classical and quantum models. The Einstein-Podolsky-Rosen-Bohm gedanken experiment and a macroscopic quantum coherence experiment proposed by Leggett and Garg are both shown to obey the extended Boole-Bell inequalities. These examples as well as additional discussions also provide reasons for apparent violations of these inequalities.