  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.
