Mutual Information is not a Reliable Measure for Variations in Total Correlations

TL;DR

This paper shows that quantum mutual information and correlation matrix norm often give different indications of correlations, highlighting the need for more comprehensive measures to accurately understand correlations in quantum systems.

Contribution

It demonstrates the discrepancy between mutual information and correlation matrix norm as measures of total correlations, providing analytic formulas and emphasizing the importance of full correlation matrix analysis.

Findings

Mutual information and correlation matrix norm differ significantly except at extreme cases.

Analytic formulas for the time derivatives of these measures are derived.

Neither measure fully captures the physics of heat exchange related to correlations.

Abstract

Correlations disguised in various forms underlie a host of important phenomena in classical and quantum systems, such as information and energy exchanges. The quantum mutual information and the norm of the correlation matrix are both considered as proper measures of total correlations. We demonstrate that, when applied to the same system, these two measures can actually show significantly different behavior except at least in two limiting cases: when there are no correlations and when there is maximal quantum entanglement. We further quantify the discrepancy by providing analytic formulas for time derivatives of the measures for an interacting bipartite system evolving unitarily. We argue that to properly account for correlations, one should consider the full information provided by the correlation matrix (and reduced states of the subsystems). Scalar quantities such as the norm of the correlation matrix or the quantum mutual information can only capture a part of the complex features of correlations. As a concrete example, we show that in describing heat exchange associated with correlations, neither of these quantities can fully capture the underlying physics.