Quasi-Bell inequalities from symmetrized products of noncommuting qubit observables

TL;DR

This paper introduces Quasi-Bell inequalities derived from symmetrized products of noncommuting qubit observables, revealing higher quantum violations than traditional Bell inequalities and emphasizing noncommutativity's role in quantum nonlocality.

Contribution

It develops a method to construct Quasi-Bell inequalities using symmetrized products of noncommuting observables, extending Bell inequalities to include noncommuting operators.

Findings

Quasi-Bell inequalities can be violated by quantum mechanics with a factor up to 3/2.

Symmetrization via the Moyal characteristic function justifies the quantum operator construction.

Higher quantum violations highlight the significance of noncommutativity in quantum nonlocality.

Abstract

Noncommuting observables cannot be simultaneously measured, however, under local hidden variable models, they must simultaneously hold premeasurement values, implying the existence of a joint probability distribution. We study the joint distributions of noncommuting observables on qubits, with possible criteria of positivity and the Fr\'echet bounds limiting the joint probabilities, concluding that the latter may be negative. We use symmetrization, justified heuristically and then more carefully via the Moyal characteristic function, to find the quantum operator corresponding to the product of noncommuting observables. This is then used to construct Quasi-Bell inequalities, Bell inequalities containing products of noncommuting observables, on two qubits. These inequalities place limits on local hidden variable models that define joint probabilities for noncommuting observables. We find Quasi-Bell inequalities have a quantum to classical violation as high as $\frac{3}{2}$, higher than conventional Bell inequalities. The result demonstrates the theoretical importance of noncommutativity in the nonlocality of quantum mechanics, and provides an insightful generalization of Bell inequalities.