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arXiv:1004.0434 · Physical Review A

Witnessing quantum discord in 2×N systems

Bogna Bylicka, Dariusz Chruściński

Published Jun 2, 2010• 96 citations
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Abstract

Bipartite states with vanishing quantum discord are necessarily separable and hence positive partial transpose (PPT). We show that $2\ifmmode\times\else\texttimes\fi{}N$ states satisfy additional property: the positivity of their partial transposition is recognized with respect to the canonical factorization of the original density operator. We call such states strong PPT (SPPT). Therefore, we provide a natural witness for a quantum discord: if a $2\ifmmode\times\else\texttimes\fi{}N$ state is not SPPT it must contain nonclassical correlations measured by quantum discord. It is an analog of the celebrated Peres-Horodecki criterion: if a state is not PPT it must be entangled.

DOI
10.1103/physreva.81.062102

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•Feb 1, 2017•1 citation
Bipartite states with vanishing quantum discord are necessarily separable and hence positive partial transpose (PPT). We show that 2\ifmmode×\else\texttimes\fiN2\ifmmode\times\else\texttimes\fi{}N2\ifmmode×\else\texttimes\fiN states satisfy additional property: the positivity of their partial transposition is recognized with respect to the canonical factorization of the original density operator. We call such states strong PPT (SPPT). Therefore, we provide a natural witness for a quantum discord: if a 2\ifmmode×\else\texttimes\fiN2\ifmmode\times\else\texttimes\fi{}N2\ifmmode×\else\texttimes\fiN state is not SPPT it must contain nonclassical correlations measured by quantum discord. It is an analog of the celebrated Peres-Horodecki criterion: if a state is not PPT it must be entangled.
Quantum correlation is a key to our understanding of quantum physics. In particular, it is essential for the powerful applications to quantum information and quantum computation. There exist quantum correlations beyond entanglement, such as quantum discord (QD) and measurement-induced nonlocality (MiN) [Phys. Rev. Lett. \textbf{106}, 120401(2011)]. In [Phys. Rev. A \textbf{77}, 022113(2008)], a subclass of PPT states so-called strong positive partial transposition (SPPT) states was introduced and it was conjectured there that SPPT states are separable. However, it was illustrated with examples in [Phys. Rev. A \textbf{81}, 064101(2010)] that this conjecture is not true. Viewing the original SPPT as SPPT up to part B, in the present paper, we define SPPT state up to part A and B respectively and present a separable class of SPPT states, that is the super SPPT (SSPPT) states, in terms of local commutativity. In addition, classical-quantum (CQ) states and nullity of MiN are characterized via local commutativity. Based on CQ states, the geometric measure of quantum discord (GMQD) for infinite-dimensional case is proposed. Consequently, we highlight the relation among MiN, QD(GMQD), SSPPT and separability through a unified approach for both finite- and infinite-dimensional systems: zero MiN implies zero QD(GMQD), zero QD(GMQD) signals SSPPT and SSPPT guarantees separability, but the converses are not valid.
We generalize the definition of strong positive partial transpose (SPPT) to the multipartite system. The tripartite case was first considered by Yu and Zhao [Int. J. Theor. Phys. 54, 292 (2014)]. In this extension, unfortunately, desired properties such as the PPT of SPPT states and the separability of super and pure SPPT states are not preserved. In contrast, this paper provides an alternative generalization to multipartite cases with these properties preserved. We also provide sufficient conditions for the separability of SPPT states.
We show that all $2\ensuremath{\bigotimes}4$ states with strong positive partial transposes (SPPTs) are separable. We also construct a family of $2\ensuremath{\bigotimes}5$ entangled SPPT states, so the conjecture on the separability of SPPT states is completely settled. In addition, we clarify the relation between the set of all $2\ensuremath{\bigotimes}d$ separable states and the set of all $2\ensuremath{\bigotimes}d$ SPPT states for the case of d=3,4d=3,4d=3,4.
A bipartite subspace SSS is called strongly positive-partial-transpose-unextendible (PPT-unextendible) if for every positive integer kkk, there is no PPT state supporting on the orthogonal complement of S⊗kS^{\otimes k}S⊗k. We show that a subspace is strongly PPT-unextendible if it contains a PPT-definite operator (a positive semidefinite operator whose partial transpose is positive definite). Applying this result, we demonstrate a simple criterion for verifying whether a set of bipartite quantum states is asymptotically indistinguishable by operations completely preserving positivity of partial transpose. This criterion is based on the observation that if the support of some bipartite state is strongly PPT-unextendible, then any set of bipartite quantum states which contains this state cannot be asymptotically distinguished by PPT operations. Utilizing this criterion, we further show that any entangled pure state and its orthogonal complement cannot be distinguished asymptotically by PPT operations. Meanwhile, we also investigate that the minimum dimension of strongly PPT-unextendible subspaces in a m⊗nm\otimes nm⊗n system is m+n−1m+n-1m+n−1, which involves an extension of the result that non-positive-partial-transpose (NPT) subspaces can be as large as any entangled subspace [N. Johnston, Phys. Rev. A 87: 064302 (2013)].