A Few Steps More Towards NPT Bound Entanglement
Łukasz Pańkowski, Marco Piani, Michał Horodecki, Paweł Horodecki · IEEE Transactions on Information Theory · 2010
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 projectorQwith Schmidt rank two states does not exceed 1/2 (called thehalf-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 forallSchmidt rank two states is bounded from above bycA⊗I+I⊗BwithA,Btraceless 4 × 4 matrices, and TrAfA+ TrBfB=1/4.