Quantum information theory and Fourier multipliers on quantum groups

Abstract

In this paper, we compute the exact value of the minimum output entropy of all quantum channels induced by Fourier multipliers acting on an arbitrary finite quantum group G. Moreover, we show that this quantity is equal to the (normalized) completely bounded minimal entropy. We also give a value for the classical capacity and the quantum capacity. 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. Contents 1 Introduction 1 2 Preliminaries 4 3 Convolutions by elements of L p (G) and c.b. maps from L 1 (G) into L p (G) 8 4 Bounded versus complete. bounded Fourier multipliers from L 1 (G) into L p (G) 10 5 Bounded versus completely bounded operators from L 1 (M) into L p (M) 16 6 ℓ p (S p d )-summing Fourier multipliers on L ∞ (G) 20 7 Entropies and capacities 22 8 Appendix: output minimal entropy as derivative of operator norms 23 Bibliography 25

Spectral triples, Coulhon-Varopoulos dimension and heat kernel estimates

“…Indeed, by a well-known interpolation argument [GJP17, p. 896], we deduce that T t cb,L 1 (VN(G))→L 2 (VN(G)) < t < 1. Now, for any t > 0, by a result of[Arh21] we obtain ϕ t 2 The weak* continuity of the semigroup (T t ) t 0 on the von Neumann algebra VN(G) implies that for any s ∈ G we have ϕ t (s) − −− → −t|•| is a (continuous) positive definite function. Recall that by a combination of [Dix77, Proposition 18.3.5 p. 357] and [Dix77, Proposition 18.3.6 p. 358] a locally compact group G is amenable if and only if the function 1 is the uniform limit over every compact set of square-integrable continuous positive-definite functions 1 .…”