Quantum information theory and Fourier multipliers on quantum groups

In this paper, we compute the exact values of the minimum output entropy and the completely bounded minimal entropy of all quantum channels induced by Fourier multipliers acting on an arbitrary finite quantum group $\mathbb{G}$. We also give a sharp upper bound of the classical capacity. Our results rely on a new and precise description of bounded Fourier multipliers from $\mathrm{L}^1(\mathbb{G})$ into $\mathrm{L}^p(\mathbb{G})$ for $1 < p \leq \infty$ where $\mathbb{G}$ 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.