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arXiv:1804.10923 · Physical Review A

Separability of multipartite quantum states with strong positive partial transpose

Lilong Qian

Published Jul 9, 2018• 9 citations
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Abstract

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.

DOI
10.1103/physreva.98.012307

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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 (PPT) unextendible if for every positive integer kkk, there is no PPT operator supporting on the orthogonal complement of ${S}^{\ensuremath{\bigotimes}k}$. 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). Based on these, we are able to propose a simple criterion for verifying whether a set of bipartite orthogonal quantum states is indistinguishable by PPT operations in the many-copy scenario. Utilizing this criterion, we further point out that any entangled pure state and its orthogonal complement cannot be distinguished by PPT operations in the many-copy scenario. On the other hand, we investigate that the minimum dimension of strongly PPT-unextendible subspaces in an $m\ensuremath{\bigotimes}n$ system is $m+n\ensuremath{-}1$, which involves a generalization of the result that non-positive-partial-transpose subspaces can be as large as any entangled subspace [N. Johnston, Phys. Rev. A 87, 064302 (2013)].