Correlation Distance and Bounds for Mutual Information

TL;DR

This paper explores the correlation distance as a measure of independence in classical and quantum systems, providing tight bounds for mutual information that outperform existing inequalities and offering new criteria for entanglement detection.

Contribution

It introduces tight lower bounds for mutual information based on correlation distance, applicable to classical variables and quantum qubits, and links these bounds to entanglement criteria.

Findings

Bounds are stronger than Pinsker inequality for mutual information.

Entangled qubits can have lower mutual information than classical variables with same correlation distance.

Correlation distance serves as a direct entanglement criterion.

Abstract

The correlation distance quantifies the statistical independence of two classical or quantum systems, via the distance from their joint state to the product of the marginal states. Tight lower bounds are given for the mutual information between pairs of two-valued classical variables and quantum qubits, in terms of the corresponding classical and quantum correlation distances. These bounds are stronger than the Pinsker inequality (and refinements thereof) for relative entropy. The classical lower bound may be used to quantify properties of statistical models that violate Bell inequalities. Entangled qubits can have a lower mutual information than can any two-valued classical variables having the same correlation distance. The qubit correlation distance also provides a direct entanglement criterion, related to the spin covariance matrix. Connections of results with classically-correlated quantum states are briefly discussed.