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.

翻译：本文研究了冯·诺依曼代数 $\mathcal{M}$ 上保持其冯·诺依曼子代数 $\mathcal{N}$ 的量子通道，即 $\mathcal{N}$-$\mathcal{N}$-双模单位完全正映射。通过引入双模量子通道的相对不可约性，我们证明了其模为 1 的特征值构成一个有限循环群，称为其相位群。此外，相应的特征空间是可逆的 $\mathcal{N}$-$\mathcal{N}$-双模，这编码了相位群的范畴化。当 $\mathcal{N}\subset \mathcal{M}$ 是 II$_1$ 型有限指标不可约子因子时，我们证明任何双模量子通道相对于其不动点的中间子因子都是相对不可约的。此外，我们可以利用量子傅里叶分析的方法，在子因子平面代数中内在地重新表述并证明这些结果，而无需引用子因子本身。