Quantum information theory and Fourier multipliers on quantum groups

08/27/2020

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by   Cédric Arhancet, et al.

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In this paper, we compute the exact value of the classical capacity and of the quantum capacity of all quantum channels induced by Fourier multipliers acting on an arbitrary finite quantum group 𝔾. We also determine the value of the minimum output entropy. Moreover, we show that this quantity is equal to the completely bounded minimal entropy. Our results rely on a new and precise description of bounded Fourier multipliers from L^1(𝔾) into L^p(𝔾) for 1 < p ≤∞ where 𝔾 is a co-amenable compact quantum group of Kac type and on the automatic completely boundedness of these multipliers that this description entails. This result is new even in the case of a group von Neumann algebra associated to an amenable discrete group.