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

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.