A few steps more towards NPT bound entanglement

  • Authors:

  • Łukasz Pankowski;Marco Piani;Michał Horodecki;Paweł Horodecki

  • Affiliations:

  • Institute of Informatics, University of Gdańsk, Gdańsk, Poland and Institute of Theoretical Physics and Astrophysics;Institute of Theoretical Physics and Astrophysics, University of Gdańsk, Gdańsk, Poland and Institute for Quantum Computing and the Department of Physics and Astronomy, University of Wat ...;Institute of Theoretical Physics and Astrophysics, University of Gdańsk, Gdańsk, Poland;Faculty of Applied Physics and Mathematics, Gdańsk University of Technology, Gdańsk, Poland

  • Venue:
  • IEEE Transactions on Information Theory
  • Year:
  • 2010

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Abstract

In this paper, existence of bound entangled states with nonpositive partial transpose (NPT) is considered. As one knows, existence of such states would in particular imply nonadditivity of distillable entanglement. Moreover, it would rule out a simple mathematical description of the set of distillable states. The particular state, known to be 1-copy nondistillable and supposed to be bound entangled, is considered. The problem of its two-copy distillability, which boils down to show that maximal overlap of some projector Q with Schmidt rank two states does not exceed 1/2 (called the half-property), is studied. First, it is shown that the maximum overlap can be attained on vectors that are not of the simple product form with respect to cut between two copies. Then, the problem in attacked twofold way: a) the half-property is proved for some wide classes of Schmidt rank two states; b) the overlap for all Schmidt rank two states is bounded from above by c A ⊗ I + I ⊗ B with A, B traceless 4 × 4 matrices, and TrA+ A + TrB+ B = 1/4.