Phase Group Categories of Bimodule Quantum Channels

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.