  We expand the set of initial states of a system and its environment that are
known to guarantee completely positive reduced dynamics for the system when the
combined state evolves unitarily. We characterize the correlations in the
initial state in terms of its quantum discord [H. Ollivier and W. H. Zurek,
Phys. Rev. Lett. 88, 017901 (2001)]. We prove that initial states that have
only classical correlations lead to completely positive reduced dynamics. The
induced maps can be not completely positive when quantum correlations
including, but not limited to, entanglement are present. We outline the
implications of our results to quantum process tomography experiments.
