Entanglement is not a lower bound for geometric discord

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.

v3: Title changed, matrix theoretic proof removed due to a gap; Supplementary Material will be added later; accepted in PRA as Rapid Communication

References in corpus (7)

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Cited by in corpus (12)

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• Geometric quantum discord through the Schatten 1-norm
• Negative eigenvalues of partial transposition of arbitrary bipartite states
• Non-Positive Partial Transpose Subspaces Can be as Large as Any Entangled Subspace
• The geometric approach to quantum correlations: Computability versus reliability
• Qubit-qudit states with positive partial transpose
• Comment on "Witnessed entanglement and the geometric measure of quantum discord"
• Best Separable Approximation of multipartite diagonal symmetric states
• Geometric discord and Measurement-induced nonlocality for well known bound entangled states
• Local Quantum Uncertainty and Bounds on Quantumness for Orthogonally invariant class of states
• Quantum and classical correlations in high temperature dynamics of two coupled large spins
• Comparing Geometric Discord and Negativity for Bipartite States