Distillability and PPT entanglement of low-rank quantum states

TL;DR

This paper advances understanding of quantum entanglement by proving distillability conditions for low-rank states, solving the rank-4 separability problem, and characterizing PPT and NPT states.

Contribution

It extends distillability results to NPT states with maximal rank and fully characterizes separability for bipartite states of rank 4.

Findings

NPT states with rank equal to the maximum of local ranks are distillable.

Bipartite rank-4 states are separable iff PPT and contain a product state in their range.

Checkerboard states are distillable iff they are NPT.

Abstract

It is known that he bipartite quantum states, with rank strictly smaller than the maximum of the ranks of its two reduced states, are distillable by local operations and classical communication. Our first main result is that this is also true for NPT states with rank equal to this maximum. (A state is PPT if the partial transpose of its density matrix is positive semidefinite, and otherwise it is NPT.) This was conjectured first in 1999 in the special case when the two local ranks are equal. Our second main result provides a complete solution of the separability problem for bipartite states of rank 4. Namely, we show that such a state is separable if and only if it is PPT and its range contains at least one product state. We also prove that the so called checkerboard states are distillable if and only if they are NPT.