On the distillablity conjecture in matrix theory

TL;DR

This paper advances understanding of the distillability conjecture in quantum information by proving two special cases involving 4x4 matrices and proposing a simplified conjecture for matrices with distinct eigenvalues.

Contribution

It proves two specific cases of the distillability conjecture for 4x4 matrices and introduces a simplified version of the conjecture for matrices with distinct eigenvalues.

Findings

Proved the first case when matrices are unitarily equivalent to block diagonal matrices.

Proved the second case when B is unitarily equivalent to -A or its transpose.

Proposed a simplified conjecture for matrices with distinct eigenvalues.

Abstract

The distillability conjecture of two-copy 4 by 4 Werner states is one of the main open problems in quantum information. We prove two special cases of the conjecture. The first case occurs when two 4 by 4 matrices A, B are both unitarily equivalent to block diagonal matrices with 2 by 2 blocks. The second case occurs when B is unitarily equivalent to either -A or the transpose of -A. Plus, we propose a simplified version of the distillability conjecture when both A and B are matrices with distinct eigenvalues.