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：本文利用量子态时间演化理论，定义了与量子过程$(\rho,\mathcal{E})$相关的熵函数$S(\rho,\mathcal{E})$，其中$\rho$为量子态，$\mathcal{E}$为描述该量子态动力学演化的量子信道。该熵函数在$S(\rho,\mathrm{id})=S(\rho)$（$\mathrm{id}$表示恒等信道）的意义上推广了冯·诺伊曼熵，且作为二分态量子联合熵的动力学对应。基于此熵函数，我们进一步定义了量子条件熵与量子互信息的动力学表达式，并证明这些信息度量满足量子熵贝叶斯法则等一系列优良性质。同时，我们运用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，从而建立了量子过程信息守恒的精确表述。