On a matrix inequality related to the distillability problem

We investigate the distillability problem in quantum information in ${\text{bbC}}^d\text{ox}{\text{bbC}}^d$. A special case of the problem has been reduced to proving a matrix inequality when $d=4$. We investigate the inequality for two families of non-normal matrices. We prove the inequality for the first family with $d=4$ and two special cases of the second family with $d\ge 4$. We also prove the inequality for all normal matrices with $d>4$.