  The reduced dynamics of an open quantum system $S$, interacting with its
environment $E$, is not completely positive, in general. In this paper, we
demonstrate that if the two following conditions are satisfied, simultaneously,
then the reduced dynamics is completely positive: (1) the reduced dynamics of
the system is linear, for arbitrary system-environment unitary evolution $U$;
and (2) the reduced dynamics of the system is linear, for arbitrary initial
state of the system $\rho_S$.
