  Two perturbation methods for the non-Markovian quantum state diffusion
(NMQSD) equation are investigated. The first perturbation method under
investigation is based on a functional expansion of the NMQSD equation, while
the second one expands the NMQSD equation in terms of the coupling strength. We
have compared the advantages of the two methods based on bipartite systems
where the accuracy of both perturbation methods can be examined by comparing
the approximations with the exact solutions. Additionally, we provide an
analytical solution for a special family of system's initial states, and the
entanglement dynamics is discussed based on this solution.
