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

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.