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Physical Review A

Entangled states with strong positive partial transpose

Kil-Chan Ha

Published Jun 17, 2010• 13 citations
Publisher

Abstract

Chru\ifmmode \acute{s}\else \'{s}\fi{}ci\ifmmode \acute{n}\else \'{n}\fi{}ski, Jurkowski, and Kossakowski [Phys. Rev. A 77, 022113 (2008)] studied quantum states with strong positive partial transpose (SPPT) and conjectured that all SPPT states are separable. We construct a two-parameter class of $3\ensuremath{\bigotimes}3$ entangled SPPT states, so the conjecture does not hold true for general SPPT states.

DOI
10.1103/physreva.81.064101

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Chru\ifmmode \acute{s}\else '{s}\fi{}ci\ifmmode \acute{n}\else '{n}\fi{}ski, Jurkowski, and Kossakowski [Phys. Rev. A 77, 022113 (2008)] studied quantum states with strong positive partial transpose (SPPT) and conjectured that all SPPT states are separable. We construct a two-parameter class of 3\ensuremath⨂33\ensuremath{\bigotimes}33\ensuremath⨂3 entangled SPPT states, so the conjecture does not hold true for general 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.
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.
Entangled states with a positive partial transpose (so-called PPT states) are central to many interesting problems in quantum theory. On the one hand, they are considered to be weakly entangled, since no pure state entanglement can be distilled from them. On the other hand, it has been shown recently that some of these PPT states exhibit genuinely high-dimensional entanglement, i.e., they have a high Schmidt number. Here we investigate the ($d\ifmmode\times\else\texttimes\fi{}d$)-dimensional PPT states for $d\ensuremath{\ge}4$ discussed recently by E. Sindici and M. Piani [Phys. Rev. A 97, 032319 (2018)]. By generalizing their methods to the calculation of Schmidt numbers, we show that a linear d/2d/2d/2 scaling of its Schmidt number in the local dimension ddd can be attained.
It is well known that the positive partial transpose (PPT) criterion for determining separability is an operational separability criterion. However, in a high-dimensional (>6) situation, this criterion is not sufficient. How can we judge whether an entangled state in any high-dimensional quantum system is PPT or not? Here we propose a linear algebra method for checking the positivity of the partial transpose of a state and present a set of entangled mixed states with non-positive partial transpose.