Answer to a question by A. Mandarino, T. Linowski and K. \.{Z}yczkowski

TL;DR

This paper investigates whether permuted powers of Haar unitary matrices are asymptotically free, providing techniques that affirmatively answer a question posed by Mandarino, Linowski, and Zyczkowski.

Contribution

It introduces methods to analyze asymptotic freeness of permuted matrix families, confirming the conjecture for the specific case studied.

Findings

Permuted powers of Haar unitary matrices are asymptotically free.

Developed techniques for analyzing permutation-induced matrix transformations.

Confirmed the open question affirmatively for the studied case.

Abstract

A recent work by A. Mandarino, T. Linowski and K. \.{Z}yczkowski left open the following question. If $ \mu_N $ is a certain permutation of entries of a $ N^2 \times N^2 $ matrix ("mixing map") and $ U_N $ is a $ N^2 \times N^2 $ Haar unitary random matrix, then is the family $ U_N, U_N^{\mu_N}, ( U_N^2 )^{\mu_N}, \dots , ( U_N^m)^{\mu_N} $ asymptotically free? (here by $A^{ \mu}$ we understand the matrix resulted by permuting the entries of $ A $ according to the permutation $ \mu $). This paper presents some techniques for approaching such problems. In particular, one easy consequence of the main result is that the question above has an affirmative answer.