  Consider the set $\mathcal{S}=\lbrace\rho_{SE}\rbrace$ of possible initial
states of the system-environment. The map which assigns to each $\rho_{S}\in
\mathrm{Tr}_{E}\mathcal{S}$ a $\rho_{SE}\in \mathcal{S}$ is called the
assignment map. The assignment map is Hermitian, in general. In this paper, we
restrict ourselves to the case that the assignment map is, in addition,
positive and show that this implies that the so-called reference state is a
Markov state. Markovianity of the reference state leads to existence of another
assignment map which is completely positive. So, the reduced dynamics of the
system is also completely positive. As a consequence, when the system $S$ is a
qubit, we show that if $\mathcal{S}$ includes entangled states, then either the
reduced dynamics is not given by a map, for, at least, one unitary time
evolution of the system-environment $U$, or the reduced dynamics is
non-positive, for, at least, one $U$.
