Geometric measures of quantum correlations: characterization, quantification, and comparison by distances and operations

TL;DR

This paper compares three geometric measures of bipartite quantum correlations using different contractive distances, establishing algebraic relations and identifying measures that are fully computable, reliable, and operationally viable.

Contribution

It provides a comprehensive comparison of geometric quantum correlation measures, deriving algebraic relations and highlighting measures that are analytically computable for qubits.

Findings

Hellinger-based discord and discord of response are analytically computable for all states with a qubit subsystem.

Established algebraic relations and inequalities between different geometric quantum correlation measures.

Identified measures that are fully computable, reliable, and operationally viable.

Abstract

We investigate and compare three distinguished geometric measures of bipartite quantum correlations that have been recently introduced in the literature: the geometric discord, the measurement-induced geometric discord, and the discord of response, each one defined according to three contractive distances on the set of quantum states, namely the trace, Bures, and Hellinger distances. We establish a set of exact algebraic relations and inequalities between the different measures. In particular, we show that the geometric discord and the discord of response based on the Hellinger distance are easy to compute analytically for all quantum states whenever the reference subsystem is a qubit. These two measures thus provide the first instance of discords that are simultaneously fully computable, reliable (since they satisfy all the basic Axioms that must be obeyed by a proper measure of quantum correlations), and operationally viable (in terms of state distinguishability). We apply the general mathematical structure to determine the closest classical-quantum state of a given state and the maximally quantum-correlated states at fixed global state purity according to the different distances, as well as a necessary condition for a channel to be quantumness breaking.