Entanglement distillation in terms of a conjectured matrix inequality

Abstract

Entanglement distillation is a basic task in quantum information, and the distillable entanglement of three bipartite reduced density matrices from a tripartite pure state has been studied in [Phys. Rev. A 84, 012325 (2011)]. We extend this result to tripartite mixed states by studying a conjectured matrix inequality, namely $\mathrm{rank}({\sum }_i{R}_i\otimes S_i)\le K\mathrm{rank}({\sum }_i{R}_i^T\otimes S_i)$ holds for any bipartite matrix $M={\sum }_i{R}_i\otimes S_i$ and Schmidt rank $K$. We prove that the conjecture holds for $M$ with $K=3$ and some special $M$ with arbitrary $K$.