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Weak Multiplicativity for Random Quantum Channels

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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 \({\mathcal{N}}\) (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.

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Authors and Affiliations

  1. Centre for Quantum Information and Quantum Foundations, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, CB3 0WA, UK

    Ashley Montanaro

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  1. Ashley Montanaro
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Correspondence to Ashley Montanaro.

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Communicated by M. B. Ruskai

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Montanaro, A. Weak Multiplicativity for Random Quantum Channels. Commun. Math. Phys. 319, 535–555 (2013). https://doi.org/10.1007/s00220-013-1680-7

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