In this work, we use the theory of quantum states over time to define an entropy $S(\rho,\mathcal{E})$ associated with quantum processes $(\rho,\mathcal{E})$, where $\rho$ is a state and $\mathcal{E}$ is a quantum channel responsible for the dynamical evolution of $\rho$. The entropy $S(\rho,\mathcal{E})$ is a generalization of the von Neumann entropy in the sense that $S(\rho,\mathrm{id})=S(\rho)$ (where $\mathrm{id}$ denotes the identity channel), and is a dynamical analogue of the quantum joint entropy for bipartite states. Such an entropy is then used to define dynamical formulations of the quantum conditional entropy and quantum mutual information, and we show such information measures satisfy many desirable properties, such as a quantum entropic Bayes' rule. We also use our entropy function to quantify the information loss/gain associated with the dynamical evolution of quantum systems, which enables us to formulate a precise notion of information conservation for quantum processes.

翻译：本文利用量子态随时间演化的理论，针对量子过程$(\rho,\mathcal{E})$定义了熵$S(\rho,\mathcal{E})$，其中$\rho$表示量子态，$\mathcal{E}$为驱动该态动力学演化的量子信道。该熵$S(\rho,\mathcal{E})$是冯·诺伊曼熵的推广——当信道退化为恒等信道$\mathrm{id}$时满足$S(\rho,\mathrm{id})=S(\rho)$，同时作为两体量子联合熵的动力学对应量。以此为基础，我们进一步构建了量子条件熵和量子互信息量的动力学表述，并证明这类信息测度满足包括量子熵贝叶斯法则在内的多项理想性质。此外，通过该熵函数量化量子系统动力学演化过程中的信息损耗/增益，我们建立了量子过程信息守恒的精确数学表述。