  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.
