  The standard theoretical descriptions of the dynamics of open quantum systems
rely on the assumption that the correlations with the environment can be
neglected at some reference (initial) time. While being reasonable in specific
instances, such as when the coupling between the system and the environment is
weak or when the interaction starts at a distinguished time, the use of
initially uncorrelated states is questionable if one wants to deal with general
models, taking into account the mutual influence that the open-system and
environmental evolutions perform on each other. Here, we introduce a
perturbative method that can be applied to any microscopic modeling of the
system-environment interaction, including fully general initial correlations.
Extending the standard technique based on projection operators that single out
the relevant part of the global dynamics, we define a family of projections
adapted to a convenient decomposition of the initial state, which involves a
convex mixture of product operators with proper environmental states. This
leads us to characterize the open-system dynamics via an uncoupled system of
differential equations, which are homogeneous and whose number is limited by
the dimensionality of the open system, for any kind of initial correlations.
Our method is further illustrated by means of two cases study, for which it
reproduces the expected dynamical behavior in the long-time regime more
consistently than the standard projection technique.
