Space-time generalization of mutual information

The mutual information characterizes correlations between spatially separatedregions of a system. Yet, in experiments we often measure dynamicalcorrelations, which involve probing operators that are also separated in time.Here, we introduce a space-time generalization of mutual information which, byconstruction, satisfies several natural properties of the mutual informationand at the same time characterizes correlations across subsystems that areseparated in time. In particular, this quantity, that we call the\emph{space-time mutual information}, bounds all dynamical correlations. Weconstruct this quantity based on the idea of the quantum hypothesis testing. Asa by-product, our definition provides a transparent interpretation in terms ofan experimentally accessible setup. We draw connections with other notions inquantum information theory, such as quantum channel discrimination. Finally, westudy the behavior of the space-time mutual information in several settings andcontrast its long-time behavior in many-body localizing and thermalizingsystems.