On quantum states over time

In 2017, D. Horsman, C. Heunen, M. Pusey, J. Barrett, and R. Spekkens proved that there is no physically reasonable assignment that takes a quantum channel and an initial state and produces a joint state on the tensor product of the input and output spaces. The interpretation was that there is a clear distinction between space and time in the quantum setting that is not visible classically, where in the latter, one can freely use Bayes' theorem to go between joint states and marginals with noisy channels. In this paper, we prove that there actually is such a physically reasonable assignment, bypassing the no-go result of Horsman et al., and we illustrate that this is achievable by restricting the domain of their assignment to a domain which represents the given data more faithfully.