Geometric quantum discord with Bures distance: the qubit case

TL;DR

This paper introduces a new geometric measure of quantum discord based on Bures distance for bipartite quantum states, providing explicit formulas and applications to quantum algorithms and two-qubit states.

Contribution

It derives a formula for geometric quantum discord with Bures distance when A is a qubit, and applies it to quantum algorithms and two-qubit states with maximally mixed marginals.

Findings

Maximum discord occurs when the unitary eigenvalues are uniformly distributed.

Explicit formula for two-qubit states with maximally mixed marginals.

Closest classical states are explicitly determined.

Abstract

The minimal Bures distance of a quantum state of a bipartite system AB to the set of classical states for subsystem A defines a geometric measure of quantum discord. When A is a qubit, we show that this geometric quantum discord is given in terms of the eigenvalues of a (2 n_B) x (2 n_B) hermitian matrix, n_B being the Hilbert space dimension of the other subsystem B. As a first application, we calculate the geometric discord for the output state of the DQC1 algorithm. We find that it takes its highest value when the unitary matrix from which the algorithm computes the trace has its eigenvalues uniformly distributed on the unit circle modulo a symmetry with respect to the origin. As a second application, we derive an explicit formula for the geometric discord of two-qubit states with maximally mixed marginals and compare it with other measures of quantum correlations. We also determine the closest classical states to such two-qubit states.