  In this paper we obtain new bounds for the minimum output entropies of random
quantum channels. These bounds rely on random matrix techniques arising from
free probability theory. We then revisit the counterexamples developed by
Hayden and Winter to get violations of the additivity equalities for minimum
output R\'enyi entropies. We show that random channels obtained by randomly
coupling the input to a qubit violate the additivity of the $p$-R\'enyi
entropy. For some sequences of random quantum channels, we compute almost
surely the limit of their Schatten $S_1 \to S_p$ norms.
