Hierarchy and dynamics of trace distance correlations

TL;DR

This paper introduces trace distance-based measures of correlations in bipartite quantum states, providing analytic expressions for Bell diagonal states and comparing their hierarchy and dynamics to other metrics like relative entropy and Hilbert-Schmidt norm.

Contribution

It offers new analytic formulas for classical and total correlations using trace distance and reveals unique properties and dynamical behaviors compared to other correlation measures.

Findings

Trace distance correlations differ from other metrics in their closest uncorrelated states.

Total correlations are strictly smaller than the sum of quantum and classical correlations.

Freezing behavior of quantum correlations occurs over a larger set of states with trace distance.

Abstract

We define and analyze measures of correlations for bipartite states based on trace distance. For Bell diagonal states of two qubits, in addition to the known expression for quantum correlations using this metric, we provide analytic expressions for the classical and total correlations. The ensuing hierarchy of correlations based on trace distance is compared to the ones based on relative entropy and Hilbert-Schmidt norm. Although some common features can be found, the trace distance measure is shown to differentiate from the others in that the closest uncorrelated state to a given bipartite quantum state is not given by the product of the marginals, and further, the total correlations are strictly smaller than the sum of the quantum and classical correlations. We compare the various correlation measures in two dynamical non-Markovian models, locally applied phase-flip channels and random external fields. It is shown that the freezing behavior, observed across all known valid measures of quantum correlations for Bell diagonal states under local phase-flip channels, occurs for a larger set of starting states for the trace distance than for the other metrics.