  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.
