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.
We expand the set of initial states of a system and its environment that are known to guarantee completely positive reduced dynamics for the system when the combined state evolves unitarily. We characterize the correlations in the initial state in terms of its quantum discord [1]. We prove that initial states that have only classical correlations lead to completely positive reduced dynamics. The induced maps can be not completely positive when quantum correlations including, but not limited to, entanglement are present.
We use the Koashi-Imoto decomposition of the degrees of freedom of joint system-environment initial states to investigate the reduced dynamics. We show that a subset of joint system-environment initial states guarantees completely positive reduced dynamics if and only if the system privately owns all quantum degrees of freedom and can locally access the classical degrees of freedom, without disturbing all joint initial states in the given subset. Furthermore, we show that the quantum mutual information for such kinds of states must be independent of the quantum degrees of freedom.
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 ${\ensuremath{\rho}}_{S}$.