Relations between Markovian and non-Markovian correlations in multitime quantum processes

In the dynamics of open quantum systems, information may propagate in time through either the system or the environment, giving rise to Markovian and non-Markovian temporal correlations, respectively. However, despite their notable coexistence in most physical situations, it is not yet clear how these two quantities may limit the existence of one another. Here, we address this issue by deriving several inequalities relating the temporal correlations of general multi-time quantum processes. The dynamics are described by process tensors and the correlations are quantified by the mutual information between subsystems of their Choi states. First, we prove a set of upper bounds to the non-Markovianity of a process given the degree of Markovianity in each of its steps. This immediately implies a non-trivial maximum value for the non-Markovianity of any process, independently of its Markovianity. Finally, we obtain how the non-Markovianity limits the amount of total temporal correlations that could be present in a given process. These results show that, although any multi-time process must pay a price in total correlations to have a given amount of non-Markovianity, this price vanishes exponentially with the number of steps of the process, while the maximum non-Markovianity grows only linearly. This implies that even a highly non-Markovian process might be arbitrarily close to having maximum total correlations if it has a sufficiently large number of steps.