Asymptotic Freeness of Unitary Matrices in Tensor Product Spaces for Invariant States

TL;DR

This paper extends the understanding of asymptotic freeness of tensor products of Haar unitary matrices, showing it holds for a broader class of states and under more general group actions, using traffic probability techniques.

Contribution

It proves asymptotic freeness for tensor products of Haar unitaries with respect to larger classes of states and general group representations, expanding prior results.

Findings

Asymptotic freeness holds for a broader class of states.

Results extend to general sequences of unitary group representations.

Freeness established under simultaneous actions of symmetric and free groups.

Abstract

In this paper, we pursue our study of asymptotic properties of families of random matrices that have a tensor structure. In previous work, the first- and second-named authors provided conditions under which tensor products of unitary random matrices are asymptotically free with respect to the normalized trace. Here, we extend this result by proving that asymptotic freeness of tensor products of Haar unitary matrices holds with respect to a significantly larger class of states. Our result relies on invariance under the symmetric group, and therefore on traffic probability. As a byproduct, we explore two additional generalisations: (i) we state results of freeness in a context of general sequences of representations of the unitary group -- the fundamental representation being a particular case that corresponds to the classical asymptotic freeness result for Haar unitary matrices, and (ii) we consider actions of the symmetric group and the free group simultaneously and obtain a result of asymptotic freeness in this context as well.