Geometric quantum discord through the Schatten 1-norm

TL;DR

This paper proposes using the Schatten 1-norm to define a consistent geometric quantum discord measure, providing an analytical expression for two-qubit Bell-diagonal states and analyzing its properties.

Contribution

It demonstrates that the Schatten 1-norm is the only p-norm suitable for defining a valid geometric quantum discord, and derives an analytical formula for Bell-diagonal states.

Findings

The 1-norm geometric discord is equivalent to negativity of quantumness.

The 1-norm is the only Schatten p-norm that yields a consistent quantum correlation measure.

The measure's monotonicity properties are analyzed.

Abstract

It has recently been pointed out that the geometric quantum discord, as defined by the Hilbert-Schmidt norm (2-norm), is not a good measure of quantum correlations, since it may increase under local reversible operations on the unmeasured subsystem. Here, we revisit the geometric discord by considering general Schatten p-norms, explicitly showing that the 1-norm is the only p-norm able to define a consistent quantum correlation measure. In addition, by restricting the optimization to the tetrahedron of two-qubit Bell-diagonal states, we provide an analytical expression for the 1-norm geometric discord, which turns out to be equivalent to the negativity of quantumness. We illustrate the measure by analyzing its monotonicity properties.