Divisibility hierarchy of open quantum systems

In the theory of open quantum systems, divisibility of the system dynamical maps is related to memory effects in the dynamics. By decomposing the system Hilbert space as a direct sum of several Hilbert spaces, we study the relationship among the corresponding dynamical maps. It is shown that if the dynamical maps of the open system possess a chain of invariant subspaces, there exists a divisibility hierarchy for their corresponding dynamics. Two classes of examples are given for illustrating these hierarchical structures. One is the pure-dephasing dynamics, and the other is the decay dynamics. Our results offer a systematic approach to obtaining the divisibility conditions and non-Markovian witnesses for these dynamics. Moreover, as a new way of decomposing open quantum systems, it is worthy of further study.