On the partial transpose of a Haar unitary matrix

Abstract

We consider the effect of a partial transpose on the limit ∗-distribution of a Haar distributed random unitary matrix. If we fix the number of blocks, b, we show that the partial transpose can be decomposed along diagonals into a sum of b matrices which are asymptotically free and identically distributed. We then consider the joint effect of different block decompositions and show that under some mild assumptions we also get asymptotic freeness.