Weak-coupling master equation for arbitrary initial conditions

The structure of the initial system-environment state is fundamental to determining the nature and characteristics of the evolution of such an open quantum system. The usual assumption is to consider that the initial system-environment state is separable. Here, we go beyond this simple case and derive the evolution equations, up to second order in a weak-coupling expansion, that describe the evolution of the reduced density matrix of the system for any arbitrary system-environment initial state. The structure of these equations allows us to determine the initial conditions for which the evolution of the reduced density matrix can be written in terms of a set of Lindblad-like equations, once considering the Markov and secular approximations. Moreover, we show that for initial states belonging to a subset of separable states such Lindblad-like equations become actual Lindblad equations and that the evolution of rho(s) thereby obtained is also trace and positive preserving.