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=US⊗UE, then we can choose
S=D, where
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
ρSE. 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
S must be chosen as a proper subset of
D to achieve linear reduced dynamics. As a byproduct, considering a convex set of possible initial states of the system-environment
S such that
TrE S=TrE D, we show that when the reduced dynamics of the system, for one system-environment unitary evolution
U1, is positive, but not completely positive, this implies that reduced dynamics is not linear for another
U2.
The reduced dynamics of an open quantum system
S, interacting with its environment
E, is not completely positive, in general. In this paper, we demonstrate that if the two following conditions are satisfied, simultaneously, then the reduced dynamics is completely positive: (1) the reduced dynamics of the system is linear, for arbitrary system-environment unitary evolution
U; and (2) the reduced dynamics of the system is linear, for arbitrary initial state of the system ${\ensuremath{\rho}}_{S}$.
It is now increasingly realized in the study of open system dynamics that initial correlations do not pose a conceptual difficulty as traditionally believed. A similar methodology as used to describe initial product states can be adopted, with the only difference being that the reduced dynamics is possibly not completely positive, entailing that only a restricted set of initial reduced states lead to a physically valid dynamics. Here we study the interplay between the initial correlations, especially in the form of highly entangled states, and the system–environment unitary. In particular, for almost any initial entangled state, one can furnish infinitely many joint unitaries that generate CP dynamics on the system. Restricting to the case of initial, pure entangled states, we obtain the scaling of the dimension of the set of these unitaries and show that it is of zero measure in the set of all possible interaction unitaries.
Consider an open quantum system S, interacting with its environment E, and also an ancillary Hilbert space R. The reduced dynamics of the system S is given by a completely positive map, if the set S={ρSE} , of possible initial states of the system-environment, can be written as a steered set from a tripartite Markov state ƮRSE. In this paper, we call such steered states as Markovian states ρSE, and study a physical case, in which the reduced dynamics of the system can be completely positive, even when the initial states of the system-environment ρSE are non-Markovian.
Starting from the Schrödinger equation for a system and a reservoir, we construct a quantum dynamical map that describes reduced dynamics. We work with an arbitrary initial state for the system and reservoir, and thus refrain from performing any factorizations. The map is found to be nonlinear, not completely positive, and not unique. To investigate the physical implications, the Bloch equations are constructed. If the freedom the new map offers us is fully exploited, then the classic inequality γ ⊥ ⩾γ ∥ /2 for the damping coefficients must be replaced by a weaker condition.