Quantum channels arising from abstract harmonic analysis

TL;DR

This paper introduces new quantum channels derived from harmonic analysis on groups, providing counterexamples to existing conjectures and exploring their properties for quantum information processing.

Contribution

It constructs novel quantum channels from harmonic analysis representations, offering counterexamples to conjectures and methods for quantum error correction.

Findings

Counterexamples to fixed point subalgebras in infinite dimensions

Counterexamples to the asymptotic quantum Birkhoff conjecture

Analysis of channel properties like quantum capacity and entanglement

Abstract

We present a new application of harmonic analysis to quantum information by constructing intriguing classes of quantum channels stemming from specific representations of multiplier algebras over locally compact groups $G$. Beginning with a representation of the measure algebra $M(G)$, we unify and elaborate on recent counter-examples to fixed point subalgebras in infinite dimensions, as well as present an application to the noiseless subsystems method of quantum error correction. Using a representation of the completely bounded Fourier multiplier algebra $McbA(G)$, we provide a new class of counter-examples to the recently solved asymptotic quantum Birkhoff conjecture, along with a systematic method of producing the examples using a geometric representation of Schur maps. Further properties of our channels including duality, quantum capacity, and entanglement preservation are discussed along with potential applications to additivity conjectures.