  A system interacting with its environment will give rise to a quantum
evolution. After tracing over the environment the net evolution of the system
can be described by a linear Hermitian map. It has recently been shown that a
necessary and sufficient condition for this evolution to be completely positive
is for the initial state to have vanishing quantum discord. In this paper, we
provide a sufficient condition for the map to be positive with respect to the
initial system-environment correlation. This could lead to ways in which to
identify positive but not completely positive maps. Illustrative examples and
suggestive procedures are also provided.
