Violation of Bell's inequality and postulate on simultaneous measurement of compatible observables

TL;DR

This paper explores the violation of Bell's inequality, critiques the postulate on simultaneous measurement of compatible observables, and proposes a new interpretation based on conditional measurement schemes, linking these to non-Kolmogorovness.

Contribution

It introduces a novel interpretation of joint probability for compatible observables, challenging traditional postulates and connecting non-Kolmogorovness with measurement schemes.

Findings

Identifies non-Kolmogorovness as a source of Bell inequality violation.

Critiques the physical validity of the simultaneous measurement postulate.

Proposes a conditional measurement interpretation for quantum observables.

Abstract

We discuss coupling of violation of Bell's inequality and non-Kolmogorovness of statistical data in the EPR-Bohm experiment. We emphasize that nonlocalty and "death of realism" are only sufficient, but not necessary conditions of non-Kolmogorovness. There can be found other sufficient conditions of non-Kolmogorovness and, hence, violation of Bell's inequality. We find one important source of non-Kolmogorovness by analyzing axiomatics of quantum mechanics. We pay attention to the postulate (due to von Neumann and Dirac) on simultaneous measurement of quantum observables given by commuting operators. This postulate is criticized as nonphysical. We propose a new interpretation of the Born-von Neumann-Dirac rule for calculation of the joint probability distribution for such observables. It gets a natural physical interpretation by considering conditional measurement scheme. We use this argument (i.e., rejection of the postulate on simultaneous measurement to motivate non-Kolmogorovness of the probabilistic structure of the EPR-Bohm experiment.