A Few Steps More Towards NPT Bound Entanglement

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 <; 3/4. Moreover, the problem has been translated into the following matrix analysis problem: bound the sum of the squares of the two largest singular values of matrix A ⊗I +I ⊗B with A,B traceless 4 × 4 matrices, and Tr A^fA + Tr B^fB = ¹/₄.