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We develop a notion of stochastic quantum trajectories. First, we construct abasis set of trajectories, called elementary trajectories, and go on to showthat any quantum dynamical process, including those that are non-Markovian, canbe expressed as a linear combination of this set. We then show that the set ofprocesses divide into two natural classes: those that can be expressed asconvex mixture of elementary trajectories and those that cannot be. The formerare shown to be entanglement breaking processes (in each step), while thelatter are dubbed coherent processes. This division of processes is analogousto 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 whilepreserving arbitrary quantum information encoded into the system. We giveexplicit expressions for the temporal correlations (quantifyingnon-Markovianity) and show that, in general, there are more quantumcorrelations than classical ones. This shows that non-Markovian quantumprocesses are indeed fundamentally different from their classical counterparts.Furthermore, we demonstrate how coherent trajectories (with the aid of coherentcontrol) could turn non-Markovianity into a resource. In the final section ofthe paper we explore this phenomenon in a geometric picture with a convenientset of basis trajectories.

Non-Markovian quantum control as coherent stochastic trajectories