Entanglement distillation in terms of Schmidt rank and matrix rank

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 $\max\{M,N\}+1$.