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 quantum channels acting on matrix algebras $\mathrm{M}_n$ belonging to an infinite number of large classes of quantum channels. We also give a upper bound of the classical capacity which is an equality in some cases. For that, we use the theory of quantum groups and 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 suitable locally compact quantum group and on the automatic completely boundedness of these multipliers that this description entails. We also connect ergodic actions of (quantum) groups to the topic of computation of the minimum output entropy and the completely bounded minimal entropy of channels. Finally, we investigate entangling breaking and $\mathrm{PPT}$ Fourier multipliers and we characterize conditional expectations which are entangling breaking.