A note on a conjecture from distillability of quantum entanglement

Abstract

A conjecture from the distillability of quantum entanglement is that when A and B are $4\times 4$ trace zero complex matrices and ${\Vert A\Vert }^2+{\Vert B\Vert }^2=1/4$ (where $\Vert \cdot \Vert$ is the Frobenius norm), the sum of squares of the largest two singular values of $A\otimes I_4+I_4\otimes B$ does not exceed 1/2. In this paper, the conjecture is proved when

• (i)
  A or B is unitarily similar to a direct sum of $2\times 2$ trace zero matrices;
• (ii)
  A and B are unitarily similar to matrices, when partitioned into $2\times 2$ blocks, having zero diagonal blocks.