Entanglement is not a lower bound for geometric discord

TL;DR

This paper disproves a conjecture that geometric discord is always lower bounded by squared negativity, providing counterexamples and extending eigenvalue bounds for partial transposition in bipartite states.

Contribution

It extends eigenvalue bounds for partial transposition to higher dimensions and critically assesses and refutes a conjecture relating geometric discord and negativity.

Findings

Counterexamples found in finite-dimensional systems.

Violations occur in $2\otimes n$ systems for all $n>2$.

The $4\otimes 4$ Werner state also violates the conjecture.

Abstract

We show that partial transposition of any $2\otimes n$ state can have at most $(n-1)$ number of negative eigenvalues. This extends a decade old result of $2\otimes 2$ case by Sanpera et al [Phys. Rev. A {\bf 58}, 826 (1998)]. We then apply this result to critically assess an important conjecture recently made in [Phys. Rev. A {\bf 84}, 052110 (2011)], namely, the (normalized) geometric discord should always be lower bounded by squared negativity. This conjecture has strengthened the common belief that measures of generic quantum correlations should be more than those of entanglement. Our analysis shows that unfortunately this is not the case and we give several counterexamples to this conjecture. All the examples considered here are in finite dimensions. Surprisingly, there are counterexamples in $2\otimes n$ for any $n>2$. Coincidentally, it appears that the $4\otimes 4$ Werner state, when seen as a $2\otimes 8$ dimensional state, also violates the conjecture. This result contributes significantly to the negative side of the current ongoing debates on the defining notion of geometric discord as a good measure of generic correlations.