On a matrix inequality related to the distillability problem

TL;DR

This paper explores a matrix inequality related to the quantum distillability problem, proving it for specific matrix families and dimensions, advancing understanding in quantum information theory.

Contribution

It introduces new proofs of a key matrix inequality for certain non-normal and normal matrices, extending results to higher dimensions.

Findings

Proved the inequality for the first family of non-normal matrices at d=4.

Established the inequality for two special cases of the second family at d≥4.

Confirmed the inequality for all normal matrices when d>4.

Abstract

We investigate the distillability problem in quantum information in $\bbC^d\ox\bbC^d$. A special case of the problem has been reduced to proving a matrix inequality when $d=4$. We investigate the inequality for two families of non-normal matrices. We prove the inequality for the first family with $d=4$ and two special cases of the second family with $d\ge4$. We also prove the inequality for all normal matrices with $d>4$.