Quantum information theory and Fourier multipliers on quantum groups
- URL: http://arxiv.org/abs/2008.12019v13
- Date: Mon, 8 Mar 2021 15:41:41 GMT
- Title: Quantum information theory and Fourier multipliers on quantum groups
- Authors: C\'edric Arhancet
- Abstract summary: We compute the exact values of the minimum output entropy and the completely bounded minimal entropy of quantum channels acting on matrix algebras.
Our results use a new and precise description of bounded Fourier multipliers from $mathrmL1(mathbbG)$ into $mathrmLp(mathbbG)$ for $1 p leq infty$ where $mathbbG$ is a co-amenable locally compact quantum group.
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- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- 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. Finally, we investigate entangling breaking and
$\mathrm{PPT}$ Fourier multipliers and we characterize conditional expectations
which are entangling breaking.
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