  The free contraction norm (or the (t)-norm) was introduced by Belinschi,
Collins and Nechita as a tool to compute the typical location of the collection
of singular values associated to a random subspace of the tensor product of two
Hilbert spaces. In turn, it was used in by them in order to obtain sharp bounds
for the violation of the additivity of the minimum output entropy for random
quantum channels with Bell states. This free contraction norm, however, is
difficult to compute explicitly. The purpose of this note is to give a good
estimate for this norm. Our technique is based on results of super convergence
in the context of free probability theory. As an application, we give a new,
simple and conceptual proof of the violation of the additivity of the minimum
output entropy.
