We consider a bipartite quantum system S (including parties A and B), interacting with an environment E through a localized quantum dynamics FSE. We call a quantum dynamics FSE localized if, e.g. the party A is isolated from the environment and only B interacts with the environment: FSE ¼ idA FBE, where idA is the identity map on the part A and FBE 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 ES ¼ idA EB, where EB 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.

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