Comment on "Witnessed entanglement and the geometric measure of quantum discord"

TL;DR

This paper critiques a recent study on the relationship between geometric quantum discord and entanglement, providing counterexamples to challenge the previously claimed hierarchy relations.

Contribution

It identifies errors in prior work and supplies analytic counterexamples, clarifying the incorrectness of the earlier hierarchy relations between quantum discord and entanglement.

Findings

Previous hierarchy relations are incorrect.

Analytic counterexamples demonstrate violations.

Clarifies the limitations of geometric discord-entanglement relations.

Abstract

In a recent article [Phys. Rev. A 86, 024302 (2012)], the authors have derived some hierarchy relations between geometric discord and entanglement (measured by negativity and its square). We point out that these results are incorrect and give analytic counterexamples. We also discuss briefly the reason for such violations.