A few steps more towards NPT bound entanglement

TL;DR

This paper investigates the existence of NPT bound entangled states, focusing on the two-copy distillability problem, and introduces the half-property to analyze overlaps with Schmidt rank two states, providing bounds and proofs for certain classes.

Contribution

It advances understanding of NPT bound entanglement by proving the half-property for specific classes and establishing bounds on overlaps, linking the problem to matrix singular value analysis.

Findings

Proved the half-property for wide classes of states.

Bound the overlap from above by c<3/4.

Translated the problem into matrix singular value bounds.

Abstract

We consider the problem of existence of bound entangled states with non-positive partial transpose (NPT). 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. Distillability is equivalent to so called n-copy distillability for some n. We consider a particular state, known to be 1-copy nondistillable, which is supposed to be bound entangled. We study 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. Such property we call the the half-property. We first show that the maximum overlap can be attained on vectors that are not of the simple product form with respect to cut between two copies. We then attack the problem in twofold way: a) prove the half-property for some classes of Schmidt rank two states b) bound the required overlap from above for all Schmidt rank two states. We have succeeded to prove the half-property for wide classes of states, and to bound the overlap from above by c<3/4. Moreover, we translate the problem into the following matrix analysis problem: bound the sum of the squares of the two largest singular values of matrix A \otimes I + I \otimes B with A,B traceless 4x4 matrices, and Tr A^\dagger A + Tr B^\dagger B = 1/4.