Structure of states for which each localized dynamics reduces to a localized subdynamics

We consider a bipartite quantum system $S$ (including parties $A$ and $B$), interacting with an environment $E$ through a localized quantum dynamics $\mathcal{F}_{SE}$ . We call a quantum dynamics $\mathcal{F}_{SE}$ localized if, e.g., the party $A$ is isolated from the environment and only $B$ interacts with the environment: $\mathcal{F}_{SE}=id_{A}\otimes \mathcal{F}_{BE}$, where $id_{A}$ is the identity map on the part $A$ and $\mathcal{F}_{BE}$ is a completely positive (CP) map on the both $B$ and $E$. We will show that the reduced dynamics of the system is also localized as $\mathcal{E}_{S}=id_{A}\otimes \bar{\mathcal{E}}_{B}$, where $\bar{\mathcal{E}}_{B}$ is a CP map on $B$, if and only if the initial state of the system-environment is a Markov state. We then generalize this result to the two following cases: when both $A$ and $B$ interact with a same environment, and when each party interacts with its local environment.