The Influence Functional in open holography: entanglement and R\'enyi entropies

Open quantum systems are defined as ordinary unitary quantum theories coupled to a set of external degrees of freedom, which are introduced to take on the r\^ole of an unobserved environment. Here we study examples of open quantum field theories, with the aid of the so-called Feynman- Vernon Influence Functional (IF), including field theories that arise in holographic duality. We interpret the system in the presence of an IF as an open effective field theory, able to capture the effect of the unobserved environment. Our main focus is on computing R\'enyi and entanglement entropies in such systems, whose description from the IF, or "open EFT", point of view we develop in this paper. The issue of computing the entanglement-R\'enyi entropies in open quantum systems is surprisingly rich, and we point out how different prescriptions for the IF may be appropriate depending on the application of choice. A striking application of our methods concerns the fine-grained entropy of subsystems when including gravity in the setup, for example when considering the Hawking radiation emitted by black holes. In this case we show that one prescription for the IF leads to answers consistent with unitary evolution, while the other merely reproduces standard EFT results, well known to be inconsistent with unitary global evolution. We establish these results for asymptotically AdS gravity in arbitrary dimensions, and illustrate them with explicit analytical expressions for the IF in the case of matter-coupled JT gravity in two dimensions.