Covariance Bell inequalities

TL;DR

This paper introduces nonlinear covariance-based Bell inequalities, demonstrating quantum violations and their potential to serve as device-independent witnesses for shared randomness, dimension, and entropy.

Contribution

It presents a novel class of covariance Bell inequalities with nonlinearity, providing analytical bounds and applications as shared randomness witnesses.

Findings

Quantum violations of covariance Bell inequalities demonstrated.

Analytical tight bounds for local and quantum correlations derived.

Covariance Bell inequalities can serve as shared randomness witnesses.

Abstract

We introduce Bell inequalities based on covariance, one of the most common measures of correlation. Explicit examples are discussed, and violations in quantum theory are demonstrated. A crucial feature of these covariance Bell inequalities is their nonlinearity; this has nontrivial consequences for the derivation of their local bound, which is not reached by deterministic local correlations. For our simplest inequality, we derive analytically tight bounds for both local and quantum correlations. An interesting application of covariance Bell inequalities is that they can act as "shared randomness witnesses": specifically, the value of the Bell expression gives device-independent lower bounds on both the dimension and the entropy of the shared random variable in a local model.