On the Partial Transpose of a Haar Unitary Matrix

Abstract

We consider the effect of a partial transpose on the limit $\ast$-distribution of a Haar distributed random unitary matrix. If we fix, $b$, the number of blocks, we show that the partial transpose can be decomposed 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.

23 pages