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arXiv:0710.1934 · Physical Review A

Quantum states with strong positive partial transpose

Dariusz Chruściński, Jacek Jurkowski, Andrzej Kossakowski

Published Feb 25, 2008• 32 citations
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Abstract

We construct a large class of bipartite $M\ensuremath{\bigotimes}N$ quantum states which defines a proper subset of states with positive partial transposes (PPTs). Any state from this class has PPT but the positivity of its partial transposition is recognized with respect to canonical factorization of the original density operator. We propose to call elements from this class states with strong positive partial transposes (SPPTs). We conjecture that all SPPT states are separable.

DOI
10.1103/physreva.77.022113

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We construct a large class of bipartite M\ensuremath⨂NM\ensuremath{\bigotimes}NM\ensuremath⨂N quantum states which defines a proper subset of states with positive partial transposes (PPTs). Any state from this class has PPT but the positivity of its partial transposition is recognized with respect to canonical factorization of the original density operator. We propose to call elements from this class states with strong positive partial transposes (SPPTs). We conjecture that all SPPT states are separable.
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.
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.
We construct a large class of quantum $d\ensuremath{\bigotimes}d$ states which are positive under partial transposition (so called PPT states). The construction is based on certain direct sum decomposition of the total Hilbert space displaying characteristic circular structure---that is why we call them circulant states. It turns out that partial transposition maps any such decomposition into another one and hence both original density matrix and its partially transposed partner share similar cyclic properties. This class contains many well-known examples of PPT states from the literature and gives rise to a huge family of completely new states.