  In this appendix to our paper with the same title posted on arxiv we give a
quick proof of an inequality that can be substituted to Hastings's result,
quoted as Lemma 1.9 in our previous paper. Our inequality is less sharp but
also appears to apply with more general (and even matricial) coefficients. It
shows that up to a universal constant all moments of the norm of a linear
combination of the form
  $$S=\sum\nolimits_j a_j U_j \otimes \bar U_j (1-P)$$ are dominated by those
of the corresponding Gaussian sum
  $$S'=\sum\nolimits_j a_j Y_j \otimes \bar Y'_j .$$ The advantage is that $S'$
is now simply separately a Gaussian random variable with respect to the
independent Gaussian random matrices $(Y_j)$ and $(Y'_j)$, and hence is much
easier to majorize. Note we plan to incorporate this appendix into our future
publication.
