Wormholes from averaging over states

An important question about black holes is to what extent a typical pure state differs from the ensemble average. We show that this question can be answered within semiclassical gravity. We focus on the quantum deviation, which measures the fluctuations in the expectation value of an operator in an ensemble of pure states. For a large class of ensembles and observables, these fluctuations are calculated by a correlation function in the eternal black hole background, which can be reliably calculated within semi-classical gravity. This implements the idea of [arXiv:2002.02971] that wormholes can arise from averages over states rather than theories. As an application, we calculate the size of the long-time correlation function〈A(t)A(0).