Description
The framework of process tensors, also known as quantum combs, is a powerful tool to characterize non-Markovian processes in terms of only system-level quantities. However, these analyses are limited in generality due to the common digitisation of processes. Currently, there is an absence of a complete framework for arbitrary open quantum processes at continuous intervals. Therefore, despite the practical success of the traditional process tensor framework, there is a lack of operational understanding of continuous time, non-Markovian open quantum systems. This is a problem at the practical level (devices take finite time to implement control) as well as dissatisfying at the foundational level (we should be able to treat fully general stochastic processes and phenomena such as multi-time correlations in way that is commensurate with Schrödinger evolution — without resorting to digitisation).
In this work, we propose a continuous generalization of the process tensor framework using continuous matrix product states (cMPS) to extend it into the continuous-time domain. Here, we encode the time process into the spatial structure of a 1D field theory state which is naturally continuous. We demonstrate how the interplay of control and environment can be expressed as the overlap of two cMPS states. Any multi-time experiment, including analog control and continuous measurements with feedback, is describable under this picture. In other words, the effects of continuous non-Markovian phenomena can be made precise. Importantly, we show how families of discrete process tensors emerge from this formalism, thereby completing the Kolmogorov picture of quantum stochastic processes in a constructive sense.
In addition to providing insight into the nature of all device noise, from the physical to the logical level, our framework presents a new way to understand continuous multi-time phenomena by offering a compact yet fully general approach to characterize non-Markovianity in continuous settings.
