We broaden the study of circulant Quantum Markov Semigroups (QMS). First, we introduce the notions of $G$-circulant GKSL generator and $G$-circulant QMS from the circulant case, corresponding to ${\mathbb{Z}}_n$, to an arbitrary finite group $G$. Second, we show that each $G$-circulant GKSL generator has a block-diagonal representation $Q\otimes {\mathbb{𝟙}}_G$, where $Q$ is a $G$-circulant matrix determined by some $\alpha \in {\ell }_2(G)$. Denoting by $H$ the subgroup of $G$ generated by the support of $\alpha$, we prove that $Q$ has its own block-diagonal matrix representation $\tilde{Q}\otimes {\mathbb{𝟙}}_r$ where $\tilde{Q}$ is an irreducible $H$-circulant matrix and $r$ is the index of $H$ in $G$. Finally, we exploit such block representations to characterize the structure, steady states, and asymptotic evolution of $G$-circulant QMSs.

我们拓宽了循环量子马尔可夫半群 (QMS) 的研究。首先，我们引入以下概念$G$-循环GKSL发生器和$G$- 来自循环案例的循环 QMS，对应于${\mathbb{Z}}_n$, 到任意有限群$G$. 其次，我们证明每个$G$-循环 GKSL 生成器具有块对角线表示$问\otimes {\mathbb{𝟙}}_G$， 在哪里$问$是一个$G$-由某些人确定的循环矩阵$\alpha \in {\ell }_{\mathrm{2个}}(G)$. 表示为$H$的子群$G$由支持产生$\alpha$, 我们证明$问$有自己的分块对角矩阵表示$\widetilde{问}\otimes {\mathbb{𝟙}}_r$在哪里$\widetilde{问}$是不可约的$H$-循环矩阵和$r$是指数$H$在$G$. 最后，我们利用这种块表示来表征结构、稳态和渐近演化$G$- 循环质量管理体系。