On the Distillability Conjecture in Matrix Theory

The distillability conjecture of two-copy 4 \(\times\) 4 Werner states is one of the main open problems in quantum information. We prove two special cases of the conjecture by leveraging certain inequalities of eigenvalues and vectorization of matrices. The first case occurs when two 4 \(\times\) 4 matrices A and B are both unitarily equivalent to block diagonal matrices with \(\varvec{2}\) by \(\varvec{2}\) blocks. It is established by leveraging unitary invariance of singular values, decomposition of positive semi-definite matrices and some basic inequalities. The second case occurs when B is unitarily equivalent to either \(\varvec{-A}\) or \(\varvec{-A}^{\varvec{T}}\). It is established by leveraging the norm of commutators and the connection between vectorization of matrices and Kronecker product. In addition, we propose a simplified version of the distillability conjecture when both A and B are matrices with distinct eigenvalues.