Non-positive-partial-transpose quantum states of rank four are distillable

TL;DR

This paper proves that bipartite quantum states of rank four with NPT are distillable, introduces a rank five NPT state that is not 1-distillable, and conjectures its overall undistillability.

Contribution

It establishes distillability for rank four NPT states and constructs a minimal rank NPT state that is not distillable, advancing understanding of quantum state distillability.

Findings

Rank four NPT states are distillable.

Constructed a rank five NPT state that is not 1-distillable.

Conjecture that the rank five state is not distillable at all.

Abstract

We show that any bipartite quantum state of rank four is distillable, when the partial transpose of the state has at least one negative eigenvalue, i.e., the state is NPT. For this purpose we prove that if the partial transpose of a two-qutrit NPT state has two non-positive eigenvalues, then the state is distillable. We further construct a parametrized two-qutrit NPT entangled state of rank five which is not 1-distillable, and show that it is not $n$-distillable for any given $n$ when the parameter is sufficiently small. This state has the smallest rank among all 1-undistillable NPT states. We conjecture that the state is not distillable.