A matrix inequality related to the entanglement distillation problem

The pure entangled state is of vital importance in the field of quantum information. The process of asymptotically extracting pure entangled states from many copies of mixed states via local operations and classical communication is called entanglement distillation. The entanglement distillability problem, which is a long-standing open problem, asks whether such process exists. The 2-copy undistillability of $4\times4$ undistillable Werner states has been reduced to the validness of the a matrix inequality, that is, the sum of the squares of the largest two singular values of matrix $A\otimes I + I \otimes B$ does not exceed $(3d-4)/d^2$ with $A,B$ traceless $d\times d$ matrices and $||A||_F^2+||B||_F^2=1/d$ when $d=4$. The latest progress, made by {\L}.~Pankowski~ et al~[IEEE Trans. Inform. Theory, 56, 4085 (2010)], shows that this conjecture holds when both matrices $A$ and $B$ are normal. In this paper, we prove that the conjecture holds when one of matrices $A$ and $B$ is normal and the other one is arbitrary. Our work makes solid progress towards this conjecture and thus the distillability problem.