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

Random quantum channels II: Entanglement of random subspaces, Renyi entropy estimates and additivity problems