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

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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.