Quantum stochastic processes: A complete theory for non-Markovian quantum phenomena

In science, we often want to characterize dynamical processes to identify the underlying physics, predict the future states, or exercise control over the system. If the state of the system at any time depends only on the state of the system at the previous time-step and some predetermined rule then the dynamics are characterized with relative ease. For instance, the dynamics of quantum mechanical systems in isolation is described in this way. However, when a quantum system repeatedly interacts with an environment, the environment often "remembers" information about the system's past. This leads to non-Markovian processes, which depend nontrivially on the state of the system at all times during the evolution. Such dynamics can not, in general, be easily characterized using conventional techniques. Indeed, since the early days of quantum mechanics, it has been a challenge to fully describe non-Markovian processes. Here we will show, using operational tools from quantum information theory, how to fully characterize any non-Markovian process. This newly developed framework allows us to build unambiguous criteria for quantum Markov processes; extend the notion of Markov order to quantum systems; cast master equations in terms of operational elements, i.e., CPTP and higher order maps; and show that our framework constitutes the theory for quantum causal modeling. Finally, using these tools we expose non-Markovianity in IBM's five-qubit computer.