  We develop a notion of stochastic quantum trajectories. First, we construct a
basis set of trajectories, called elementary trajectories, and go on to show
that any quantum dynamical process, including those that are non-Markovian, can
be expressed as a linear combination of this set. We then show that the set of
processes divide into two natural classes: those that can be expressed as
convex mixture of elementary trajectories and those that cannot be. The former
are shown to be entanglement breaking processes (in each step), while the
latter are dubbed coherent processes. This division of processes is analogous
to separable and entangled states. In the second half of the paper, we show,
with an information theoretic game, that when a process is non-Markovian,
coherent trajectories allow for decoupling from the environment while
preserving arbitrary quantum information encoded into the system. We give
explicit expressions for the temporal correlations (quantifying
non-Markovianity) and show that, in general, there are more quantum
correlations than classical ones. This shows that non-Markovian quantum
processes are indeed fundamentally different from their classical counterparts.
Furthermore, we demonstrate how coherent trajectories (with the aid of coherent
control) could turn non-Markovianity into a resource. In the final section of
the paper we explore this phenomenon in a geometric picture with a convenient
set of basis trajectories.
