In this paper, we study the quantum channel on a von Neuamnn algebras $\mathcal{M}$ preserving a von Neumann subalgebra $\mathcal{N}$, namely $\mathcal{N}$-$\mathcal{N}$-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 relative 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 $\mathcal{N}$ 的量子信道，即 $\mathcal{N}$-$\mathcal{N}$-双模单位完全正映射。通过引入双模量子信道的相对不可约性，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，它们编码了相位群的范畴化。当 $\mathcal{N}\subset \mathcal{M}$ 是 II$_1$ 型有限指数不可约子因子时，我们证明了任何双模量子信道对其不动点的中间子因子而言都是相对不可约的。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。