Is the dynamics of open quantum systems always linear?

Abstract

We study the influence of the preparation of an open quantum system on its reduced time evolution. In contrast to the frequently considered case of an initial preparation where the total density matrix factorizes into a product of a system density matrix and a bath density matrix the time evolution generally is no longer governed by a linear map nor is this map affine. Put differently, the evolution is truly nonlinear and cannot be cast into the form of a linear map plus a term that is independent of the initial density matrix of the open quantum system. As a consequence, the inhomogeneity that emerges in formally exact generalized master equations is in fact a nonlinear term that vanishes for a factorizing initial state. The general results are elucidated with the example of two interacting spins prepared at thermal equilibrium with one spin subjected to an external field. The second spin represents the environment. The field allows the preparation of mixed density matrices of the first spin that can be represented as a convex combination of two limiting pure states, i.e. the preparable reduced density matrices make up a convex set. Moreover, the map from these reduced density matrices onto the corresponding density matrices of the total system is affine only for vanishing coupling between the spins. In general, the set of the accessible total density matrices is nonconvex.

Geometric magnetism in open quantum systems

“…(39) below. In particular, it should be noted that exact open quantum system dynamics generally is (i) neither linear [40] (ii) nor can it be described by trace preserving completely positive maps [41,42]. Attempts to resort to approximations (e.g.…”

Distance between quantum states in the presence of initial qubit-environment correlations: A comparative study

“…The Helinger distance can explicitly be calculated using the relation for the affinity (8). Then the affinity is expressed by the relation [2]…”

“…Moreover, contractivity of quantum evolution can break down provided that the system is initially correlated with its environment. Effects induced by such correlations have been studied in various context [7][8][9][10]. First experiments on initial system-environment correlations are reported in Ref.…”

Biexponential decay and ultralong coherence of a qubit

“…Unfortunately, it can violate the positivity of the density matrix and the second law of thermodynamics [56]. Alternatively, one could resort to stochastic models [57,58], non-Markovianity [59,60] or nonlinear dynamics for the reduced (open) quantum system [47,[61][62][63][64][65][66][67][68][69][70][71][72].…”