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arXiv:quant-ph/9910022 · Physical Review A

Distillability and partial transposition in bipartite systems

Wolfgang Dür, J. I. Cirac, Maciej Lewenstein, Dagmar Bruß

Published May 17, 2000• 221 citations
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Abstract

We study the distillability of a certain class of bipartite density operators which can be obtained via depolarization starting from an arbitrary one. Our results suggest that nonpositivity of the partial transpose of a density operator is not a sufficient condition for distillability, when the dimension of both subsystems is higher than 2.

DOI
10.1103/physreva.61.062313

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We study the distillability of a certain class of bipartite density operators which can be obtained via depolarization starting from an arbitrary one. Our results suggest that nonpositivity of the partial transpose of a density operator is not a sufficient condition for distillability, when the dimension of both subsystems is higher than 2.
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We introduce a formalism that connects entanglement witnesses and the distillation and activation properties of a state. We apply this formalism to two cases: First, we rederive the results presented in Eggeling et al. (e-print quant-ph/0104095), namely, that one copy of any bipartite state with nonpositive partial transpose (NPPT) is either distillable, or activatable. Second, we show that there exist three-partite NPPT states, with the property that two copies can neither be distilled, nor activated.
We analyze the separability properties of density operators supported on ${C}^{2}\ensuremath{\bigotimes}{C}^{N}$ whose partial transposes are positive operators. We show that if the rank of $\ensuremath{\rho}$ equals N then it is separable, and that bound entangled states have ranks larger than N. We also give a separability criterion for a generic density operator such that the sum of its rank and the one of its partial transpose does not exceed 3N.3N.3N. If it exceeds this number, we show that one can subtract product vectors until it decreases to 3N,3N,3N, while keeping the positivity of $\ensuremath{\rho}$ and its partial transpose. This automatically gives us a sufficient criterion for separability for general density operators. We also prove that all density operators that remain invariant after partial transposition with respect to the first system are separable.