Full and efficient characterisation of non-Markovian quantum processes

In science, we often want to characterise dynamical processes to identify the underlying physics and predict the future states of 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 characterised with relative ease. For instance, the dynamics of quantum mechanical systems in isolation is described in this way. However, when a quantum system repeatedly interact 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 its evolution. Such dynamics are not, in general, be easily characterised using conventional techniques. Indeed, since the early days of quantum mechanics it has been a challenge to describe non-Markovian processes. Here we will show, using operational tools from quantum information theory, how to fully characterise any non-Markovian process. Using this we give an unambiguous criteria for quantum Markov processes. Next, we construct a mapping from a multi-time process to a many-body state using linear (in the number of time steps) amount of bipartite entanglement. The many-body state can be measured to any desired precision, thus the process can be characterised to any desired precision.