Phase Group Categories of Bimodule Quantum Channels
Linzhe Huang, Chunlan Jiang, Zhengwei Liu, Jinsong Wu
TL;DR
The paper develops a Perron–Frobenius framework for bimodule quantum channels with symmetry given by a subalgebra $\mathcal{N}\subseteq\mathcal{M}$, extending to infinite-dimensional von Neumann settings. It encodes bimodule CP maps as elements $y_{\Phi}$ in $\mathcal{N}'\cap\mathcal{M}_1$ and uses their Fourier multipliers $\widehat{\Phi}$ in $\mathcal{M}'\cap\mathcal{M}_2$ to derive spectral structure; crucially, eigenvalues on the unit circle form a finite cyclic group $\Gamma$, and the corresponding eigenspaces are invertible $\mathcal{N}$-$\mathcal{N}$-bimodules that assemble into a unitary fusion category, providing a categorified phase group. In the finite-index irreducible II$_1$ setting, the fixed-point intermediate subfactor yields relative irreducibility of the bimodule channel, linking the phase-group analysis to subfactor theory. The authors also present intrinsic planar-algebra reformulations via quantum Fourier analysis, and extend the results to finite inclusions, offering a robust, intrinsic description that bridges operator algebras, quantum information, and planar algebras. This work thus elucidates the symmetry-encoded spectral data of bimodule quantum channels and their categorical structure, with potential implications for quantum symmetries in infinite-dimensional contexts.
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.