Quantum information theory and Fourier multipliers on quantum groups

TL;DR

This paper develops a new approach using quantum group theory to compute entropies and capacities of quantum channels, revealing connections to Fourier multipliers, quantum hypergroups, and classical channels.

Contribution

It introduces a novel method leveraging locally compact quantum groups to analyze quantum channels and their entropic properties, including a description of Fourier multipliers and their applications.

Findings

Exact values for minimum output and minimal entropy of quantum channels.

Identification of Fourier multipliers with classical quantum channels like dephasing and depolarizing channels.

Upper bounds on classical capacities that are sharp in the commutative case.

Abstract

In this paper, we compute the exact values of the minimum output entropy and the completely bounded minimal entropy of very large classes of quantum channels acting on matrix algebras $\mathrm{M}_n$. Our new and simple approach relies on the theory of locally compact quantum groups and our results use 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 locally compact quantum group and on the automatic completely boundedness of these multipliers that this description entails. Indeed, our approach even allows to use convolution operators on quantum hypergroups. This enable us to connect equally the topic of computation of entropies and capacities to subfactor planar algebras. We also give a upper bound of the classical capacity of each considered quantum channel which is already sharp in the commutative case. Quite surprisingly, we observe by direct computations that some Fourier multipliers identifies to direct sums of classical examples of quantum channels (as dephasing channel or depolarizing channels). Indeed, we show that the study of unital qubit channels can be seen as a part of the theory of Fourier multipliers on the von Neumann algebra of the quaternion group $\mathbb{Q}_8$. Unexpectedly, we also connect ergodic actions of (quantum) groups to this topic of computation, allowing some transference to other channels. We also connect the Quantum Harmonic analysis of Werner. Finally, we investigate entangling breaking and $\mathrm{PPT}$ Fourier multipliers and we characterize conditional expectations which are entangling breaking.