Decomposition and partial trace of positive matrices with Hermitian   blocks
  Let H be a positive semidefinite matrix partitioned into Hermitian blocks. Then, up to a direct sum operation, H is the average of matrices isometrically congruent to its partial trace. A few corollaries are given, related to important inequalities in quantum information theory such as the Nielsen-Kempe separability criterion. 
Jean-Christophe Bourin, Eun-Young Lee
math.FA