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 µ N is a certain permutation of entries of a N 2 × N 2 matrix ("mixing map") and U N is a N 2 × N 2 Haar unitary random matrix, then is the family U N , U µN N , (U 2 N ) µN , . . . , (U m N ) µN asymptotically free? (here by A µ we understand the matrix resulted by permuting the entries of A according to the permutation µ). 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.