Quantifying the influence of the initial state on the dynamics of an open quantum system

A small system in contact with a macroscopic environment usually approaches an asymptotic state, determined only by some macroscopic properties of the environment, such as the temperature or the chemical potential. In the long-time limit, the state of the small system is thus expected to be independent of its initial state. In some situations, however, the asymptotic state of the system is influenced by its initial state, and some information about the initial state is kept for all times. Motivated by this finding, we propose a measure to quantify the influence of the initial state of an open system on its dynamics. Using this measure we derive conditions under which the asymptotic state exists and is unique and discuss how this can be used to characterize the asymptotic behavior of an open system. We demonstrate our concepts for the dynamics of the spin-boson model, identify three qualitatively different long-time behaviors, and discuss how they can be distinguished based on the proposed measure.