  In a previous paper, we proved that the limit of the collection of possible
eigenvalues of output states of a random quantum channel is a deterministic,
compact set K_{k,t}. We also showed that the set K_{k,t} is obtained, up to an
intersection, as the unit ball of the dual of a free compression norm. In this
paper, we identify the maximum of l^p norms on the set K_{k,t} and prove that
the maximum is attained on a vector of shape (a,b,...,b) where a > b. In
particular, we compute the precise limit value of the minimum output entropy of
a single random quantum channel. As a corollary, we show that for any eps > 0,
it is possible to obtain a violation for the additivity of the minimum output
entropy for an output dimension as low as 183, and that for appropriate choice
of parameters, the violation can be as large as log 2 - eps. Conversely, our
result implies that, with probability one, one does not obtain a violation of
additivity using conjugate random quantum channels and the Bell state, in
dimension 182 and less.
