Positivity of the assignment map implies complete positivity of the reduced dynamics

Consider the set $\mathcal{S}=\lbraceρ_{SE}\rbrace$ of possible initial states of the system-environment. The map which assigns to each $ρ_{S}\in \mathrm{Tr}_{E}\mathcal{S}$ a $ρ_{SE}\in \mathcal{S}$ 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 $U$.

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