Measurement incompatibility vs. Bell non-locality: an approach via tensor norms

TL;DR

This paper explores the relationship between measurement incompatibility and Bell non-locality using tensor norms, identifying conditions under which incompatibility guarantees Bell inequality violations, with a focus on the CHSH inequality.

Contribution

It introduces a tensor norm framework to compare measurement incompatibility and Bell violation, establishing conditions for their equivalence and characterizing when incompatibility suffices for violation.

Findings

CHSH inequality uniquely satisfies the conditions for equivalence

Tensor norms effectively characterize measurement compatibility and Bell violations

Provides sufficient conditions linking measurement incompatibility to Bell inequality violations

Abstract

Measurement incompatibility and quantum non-locality are two key features of quantum theory. Violations of Bell inequalities require quantum entanglement and incompatibility of the measurements used by the two parties involved in the protocol. We analyze the converse question: for which Bell inequalities is the incompatibility of measurements enough to ensure a quantum violation? We relate the two questions by comparing two tensor norms on the space of dichotomic quantum measurements: one characterizing measurement compatibility and the second one characterizing violations of a given Bell inequality. We provide sufficient conditions for the equivalence of the two notions in terms of the matrix describing the correlation Bell inequality. We show that the CHSH inequality and its variants are the only ones satisfying it.