Weak multiplicativity for random quantum channels

TL;DR

This paper demonstrates that although random quantum channels violate strong multiplicativity, a weaker form still holds, showing exponential decay of the maximum output p-norm for multiple channel copies.

Contribution

It introduces a weaker multiplicativity property for random quantum channels and provides probabilistic bounds using random matrix theory techniques.

Findings

Maximum output p-norm decays exponentially with the number of channel copies

Weak multiplicativity holds for random quantum channels for any p>1

Operator norm bounds are derived using random matrix theory

Abstract

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 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 n copies of N decays exponentially with n. The proof is based on relaxing the maximum output infinity-norm of 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.