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Open quantum systems are defined as ordinary unitary quantum theories coupledto a set of external degrees of freedom, which are introduced to take on ther\^ole of an unobserved environment. Here we study examples of open quantumfield theories, with the aid of the so-called Feynman- Vernon InfluenceFunctional (IF), including field theories that arise in holographic duality. Weinterpret the system in the presence of an IF as an open effective fieldtheory, able to capture the effect of the unobserved environment. Our mainfocus is on computing R\'enyi and entanglement entropies in such systems, whosedescription 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 quantumsystems is surprisingly rich, and we point out how different prescriptions forthe IF may be appropriate depending on the application of choice. A strikingapplication of our methods concerns the fine-grained entropy of subsystems whenincluding gravity in the setup, for example when considering the Hawkingradiation emitted by black holes. In this case we show that one prescriptionfor the IF leads to answers consistent with unitary evolution, while the othermerely reproduces standard EFT results, well known to be inconsistent withunitary global evolution. We establish these results for asymptotically AdSgravity in arbitrary dimensions, and illustrate them with explicit analyticalexpressions for the IF in the case of matter-coupled JT gravity in twodimensions.

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