Phase Group Categories of Bimodule Quantum Channels

TL;DR

This paper introduces the concept of phase groups for bimodule quantum channels on von Neumann algebras, revealing their algebraic structure and categorification, with applications to subfactor theory and planar algebras.

Contribution

It defines the phase group for bimodule quantum channels, proves its properties, and connects it to subfactor theory and planar algebra frameworks.

Findings

Eigenvalues with modulus 1 form a finite cyclic group

Eigenspaces are invertible bimodules encoding the phase group

Results can be reformulated intrinsically in subfactor planar algebras

Abstract

In this paper, we study the quantum channel on a von Neuamnn algebra $\mathcal{M}$ preserving a von Neumann subalgebra $\mathcal{N}$, namely an $\mathcal{N}$-$\mathcal{N}$-bimodule unital completely positive map. By introducing the relative irreducibility of a bimodule quantum channel, we show that its eigenvalues with modulus 1 form a finite cyclic group, called its phase group. Moreover, the corresponding eigenspaces are invertible $\mathcal{N}$-$\mathcal{N}$-bimodules, which encode a categorification of the phase group. When $\mathcal{N}\subset \mathcal{M}$ is a finite-index irreducible subfactor of type II$_1$, we prove that any bimodule quantum channel is relatively irreducible for the intermediate subfactor of its fixed points. In addition, we can reformulate and prove these results intrinsically in subfactor planar algebras without referring to the subfactor using the methods of quantum Fourier analysis.