  We provide a general and consistent formulation for linear subsystem quantum
dynamical maps, developed from a minimal set of postulates, primary among which
is a relaxation of the usual, restrictive assumption of uncorrelated initial
system-bath states. We describe the space of possibilities admitted by this
formulation, namely that, far from being limited to only completely positive
(CP) maps, essentially any $\mathbb{C}$-linear, Hermiticity-preserving,
trace-preserving subsystem map can arise as a legitimate subsystem dynamical
map from a joint unitary evolution of a system coupled to a bath. The price
paid for this added generality is a trade-off between the set of admissible
initial states and the allowed set of joint system-bath unitary evolutions. As
an application we present a simple example of a non-CP map constructed as a
subsystem dynamical map that violates some fundamental inequalities in quantum
information theory, such as the quantum data processing inequality.
