Answer to a question by A. Mandarino, T. Linowski and K. Życzkowski

Abstract

A recent work by A. Mandarino, T. Linowski and K. Życzkowski 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.