  We show that quantum subdynamics of an open quantum system can always be
described by a Hermitian map, irrespective of the form of the initial total
system state. Since the theory of quantum error correction was developed based
on the assumption of completely positive (CP) maps, we present a generalized
theory of linear quantum error correction, which applies to any linear map
describing the open system evolution. In the physically relevant setting of
Hermitian maps, we show that the CP-map based version of quantum error
correction theory applies without modifications. However, we show that a more
general scenario is also possible, where the recovery map is Hermitian but not
CP. Since non-CP maps have non-positive matrices in their range, we provide a
geometric characterization of the positivity domain of general linear maps. In
particular, we show that this domain is convex, and that this implies a simple
algorithm for finding its boundary.
