Quantum measurement incompatibility does not imply Bell nonlocality

TL;DR

This paper demonstrates that quantum measurement incompatibility, characterized by joint measurability, does not necessarily lead to Bell inequality violations, challenging assumptions about their direct relationship in quantum nonlocality.

Contribution

The authors explicitly construct a set of non-joint measurable POVMs that cannot produce Bell nonlocality regardless of shared entanglement or Bob's measurements.

Findings

Measurement incompatibility does not imply Bell nonlocality.

Existence of non-joint measurable POVMs with local hidden variable models.

Bell violations require more than measurement incompatibility.

Abstract

We discuss the connection between the incompatibility of quantum measurements, as captured by the notion of joint measurability, and the violation of Bell inequalities. Specifically, we present explicitly a given a set of non jointly measurable POVMs $\mathcal{M}_A$ with the following property. Considering a bipartite Bell test where Alice uses $\mathcal{M}_A$, then for any possible shared entangled state $\rho$ and any set of (possibly infinitely many) POVMs $\mathcal{N}_B$ performed by Bob, the resulting statistics admits a local model, and can thus never violate any Bell inequality. This shows that quantum measurement incompatibility does not imply Bell nonlocality in general.