On the Partial Transpose of a Haar Unitary Matrix

TL;DR

This paper investigates how partial transpose operations affect the asymptotic distribution of Haar random unitary matrices, revealing conditions under which block decompositions become asymptotically free.

Contribution

It demonstrates that partial transpose decompositions of Haar unitaries are asymptotically free and explores joint effects of multiple block decompositions.

Findings

Partial transpose decompositions are asymptotically free.

Block decompositions can be combined under mild conditions.

Results apply to limit *-distributions of Haar unitaries.

Abstract

We consider the effect of a partial transpose on the limit $*$-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.