A matrix inequality related to the entanglement distillation problem

Understand

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 $d=4$.