Entanglement distillation in terms of a conjectured matrix inequality

TL;DR

This paper explores a conjectured matrix inequality related to entanglement distillation in quantum information, extending previous results to tripartite mixed states and proving the conjecture for specific cases.

Contribution

It introduces a new conjectured matrix inequality and proves its validity for certain cases, advancing understanding of entanglement distillation in quantum states.

Findings

Proved the conjecture for matrices with Schmidt rank 3.

Validated the inequality for some special matrices with arbitrary Schmidt rank.

Extended entanglement distillation results to tripartite mixed states.

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 $\mathop{\rm rank}(\sum_i R_i \otimes S_i)\leq K \mathop{\rm rank}(\sum_i R_i^T \otimes S_i)$ 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 $K$.