Entanglement distillation in terms of Schmidt rank and matrix rank

TL;DR

This paper investigates entanglement distillation for bipartite quantum states based on their Schmidt and matrix ranks, revealing conditions for distillability and equivalence to classical states.

Contribution

It characterizes distillability of bipartite states with specific Schmidt and matrix ranks, providing new criteria and equivalence conditions for entanglement distillation.

Findings

All Schmidt rank two states are locally equivalent to classical-classical states.

Schmidt rank three states are 1-undistillable.

Low-rank B-irreducible NPT states are distillable under certain conditions.

Abstract

Entanglement distillation is a key task in quantum-information processing. In this paper, we distill non-positive-partial-transpose (NPT) bipartite states of some given Schmidt rank and matrix rank. We show that all bipartite states of Schmidt rank two are locally equivalent to classical-classical states, and all bipartite states of Schmidt rank three are 1-undistillable. Subsequently, we show that low-rank B-irreducible NPT states are distillable for large-rank reduced density operators by proving low-rank B-irreducible NPT state whose range contains a product vector is distillable. Eventually, we present an equivalent condition to distill $M\times N$ bipartite states of rank $\max\{M,N\}+1$.