Deviation from linear reduced dynamics always occurs for each non-factorisable system-environment unitary evolution

Abstract

In the simplest approximation, the reduced dynamics of a quantum system $S$ interacting with its environment $E$ is considered to be given by a completely positive map. But, in general, this is not the case. In fact, the reduced dynamics of the system in not even linear, in general. Whether the reduced dynamics is linear or not is determined by two factors: the set of possible initial states of the system-environment $\mathcal{S}=\left\lbrace ρ_{SE} \right\rbrace $, and the joint system-environment unitary evolution $U$. When $U$ is factorisable as $U=U_S\otimes U_E$, then we can choose $\mathcal{S}=\mathcal{D}$, where $\mathcal{D}$ is the set of all system-environment density operators. In other words, when $U$ is factorisable, the reduced dynamics of the system $S$ is linear (in fact unitary) for arbitrary initial state of the system-environment $ρ_{SE}$. We show that this result cannot be generalized to any non-factorisable $U$: For any non-factorisable unitary evolution of the whole system-environment $U$, the set $\mathcal{S}$ must be chosen as a proper subset of $\mathcal{D}$ to achieve linear reduced dynamics. As a byproduct, considering a convex set of possible initial states of the system-environment $\mathcal{S}$ such that $\mathrm{Tr}_{E} \ \mathcal{S}=\mathrm{Tr}_{E} \ \mathcal{D}$, we show that when the reduced dynamics of the system, for one system-environment unitary evolution $U_1$, is positive, but not completely positive, this implies that reduced dynamics is not linear for another $U_2$.