In this note, we extend some results on positive and completely positive trace-preserving maps (called quantum channels in the CPTP case) from finite-dimensional to infinite-dimensional Hilbert space. Specifically, we mainly consider whether the fixed state of a quantum channel Φ on $\mathcal{T}\left(\mathscr{H}\right)$ exists, where $\mathcal{T}\left(\mathscr{H}\right)$ is the Banach algebra of all trace-class operators on the Hilbert space $\mathscr{H}$. We show that there exist the approximation states ρ_n for every quantum channel Φ. In particular, there is a quantum channel on $\mathcal{T}\left(\mathscr{H}\right)$, which has not a fixed state. Also, we get the relationship between the fixed points of $\Phi \left(\left|A\right|\right)=\left|A\right|$ and Φ(A) = ωA, where ω is the complex number with |ω| = 1 and $A\in \mathcal{T}\left(\mathscr{H}\right)$.

在本说明中，我们将一些正和完全正迹保留映射（在 CPTP 情况下称为量子通道）上的结果从有限维扩展到无限维希尔伯特空间。具体来说，我们主要考虑一个量子信道 Φ 的固定态是否在$\mathcal{吨}\left(\mathscr{H}\right)$存在，哪里$\mathcal{吨}\left(\mathscr{H}\right)$是希尔伯特空间上所有迹类算子的 Banach 代数$\mathscr{H}$. 我们表明存在每个量子通道 Φ 的近似状态ρ _{n 。}特别是，有一个量子信道$\mathcal{吨}\left(\mathscr{H}\right)$，它没有固定的状态。此外，我们得到固定点之间的关系$\Phi \left(\left|A\right|\right)=\left|A\right|$和 Φ( A ) = ωA，其中ω是具有 | 的复数 ω | = 1 和$A\in \mathcal{吨}\left(\mathscr{H}\right)$.