  We investigate completely positive maps for an open system interacting with
its environment. The families of the initial states for which the reduced
dynamics can be described by a completely positive map are identified within
the framework of direct-sum decomposition of state space. They includes not
only separable states with vanishing or nonvanishing quantum discord but also
entangled states. A general expression of the families as well as the Kraus
operators for the completely positive maps are explicitly given. It
significantly extends the previous results.
