A note on a conjecture from distillability of quantum entanglement

A conjecture from the distillability of quantum entanglement is that when A and B are 4 × 4 trace zero complex matrices and ‖ A ‖ 2 + ‖ B ‖ 2 = 1 / 4 (where ‖ ⋅ ‖ is the Frobenius norm), the sum of squares of the largest two singular values of A ⊗ I 4 + I 4 ⊗ 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 × 2 trace zero matrices; (ii) A and B are unitarily similar to matrices, when partitioned into 2 × 2 blocks, having zero diagonal blocks.