Measurements incompatible in Quantum Theory cannot be measured jointly in any other local theory

TL;DR

The paper proves that any pair of incompatible quantum observables can violate a Bell inequality, and this incompatibility persists in any no-signaling theory, establishing a fundamental link between measurement incompatibility and nonlocality.

Contribution

It demonstrates that quantum incompatibility of observables is equivalent to the potential for Bell inequality violation in any no-signaling theory, extending known results beyond qubits.

Findings

Incompatibility implies Bell violation in quantum and no-signaling theories.

The CHSH inequality is dual to joint measurability for arbitrary dimensions.

Provides a simple criterion for joint measurability beyond qubits.

Abstract

It is well known that jointly measurable observables cannot lead to a violation of any Bell inequality - independent of the state and the measurements chosen at the other site. In this letter we prove the converse: every pair of incompatible quantum observables enables the violation of a Bell inequality and therefore must remain incompatible within any other no-signaling theory. While in the case of von Neumann measurements it is sufficient to use the same pair of observables at both sites, general measurements can require different choices. The main result is obtained by showing that for arbitrary dimension the CHSH inequality provides the Lagrangian dual of the characterization of joint measurability. This leads to a simple criterion for joint measurability beyond the known qubit case.