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Weak multiplicativity for random quantum channels. (English) Zbl 1269.81027

Summary: 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 \(\infty \)-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.

MSC:

81P45 Quantum information, communication, networks (quantum-theoretic aspects)
81P15 Quantum measurement theory, state operations, state preparations
47B65 Positive linear operators and order-bounded operators
94A40 Channel models (including quantum) in information and communication theory
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