Weak Multiplicativity for Random Quantum Channels

It is known that random quantum channels exhibit significant violations of multiplicativity of maximum output p-norms for any p > 1. In this work, we show that a weaker variant of multiplicativity nevertheless holds for these channels. For any constant p > 1, given a random quantum channel ${mathcal{N}}$ (i.e. a channel whose Stinespring representation corresponds to a random subspace S), we show that with high probability the maximum output p-norm of ${mathcal{N}^{otimes n}}$ decays exponentially with n. The proof is based on relaxing the maximum output ∞-norm of ${mathcal{N}}$ to the operator norm of the partial transpose of the projector onto S, then calculating upper bounds on this quantity using ideas from random matrix theory.