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

Abstract

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: ${\mathcal{F}}_{SE}=i{d}_A\otimes {\mathcal{F}}_{BE}$, where $i{d}_A$ 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 ${\mathcal{E}}_S=i{d}_A\otimes {\bar{\mathcal{E}}}_B$, 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.

9 pages

References in corpus (3)

• Structure of states which satisfy strong subadditivity of quantum entropy with equality
• Completely Positive Maps and Classical Correlations
• Structure of correlated initial states that guarantee completely positive reduced dynamics

Cited by in corpus (3)

• Markovianity of the reference state, complete positivity of the reduced dynamics, and monotonicity of the relative entropy
• Entanglement increase from local interaction in the absence of initial quantum correlation in the environment and between the system and the environment
• Entanglement revival can occur only when the system-environment state is not a Markov state