Coarse-grained hidden entropy production in partially inaccessible quantum jump trajectories

We consider an open quantum system for which only a subset of all possible transitions are accessible, while the remaining ones are hidden from direct observation. Using a modification of the notion of quantum jump trajectories we introduce the coarse-grained hidden entropy, which quantifies the entropy production in the hidden subsystem conditioned on our observations of the visible part. The entropy production consisting of the sum of visible and coarse-grained hidden entropy is shown to satisfy an integral fluctuations theorem. Depending on the nature of the continuously occurring measurement and feedback processes between the interacting subsystems, the visible entropy can take negative values in which case the hidden systems acts as a Maxwell's demon. Our results allow us to study quantum-mechanical effects in autonomous systems, such as the autonomous quantum Maxwell's demon we introduce as an illustrative example.