Partial trace relations beyond normal matrices

We investigate the relationship between partial traces and their dilations for general complex matrices, focusing on two main aspects: the existence of (joint) dilations and norm inequalities relating partial traces and their dilations. Throughout our analysis, we pay particular attention to rank constraints. We find that every pair of matrices of equal size and trace admits dilations of any rank larger than one. We generalize Audenaert's subadditivity inequality to encompass general matrices, multiple tensor factors, and different norms. A central ingredient for this is a novel majorization relation for Kronecker sums. As an application, we extend the interval of Werner states in which they are provably 2-undistillable in any dimension $d\geq4$. We also prove new Schmidt-number witnesses and $k$-positive maps.