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The entanglement entropy associated with a spatial boundary in quantum fieldtheory is UV divergent, with the leading term proportional to the area of theboundary. For a class of quantum states defined by a path integral, theCallan-Wilczek formula gives a geometrical definition of the entanglemententropy. We show that, for this class of quantum states, the entanglemententropy is rendered UV-finite by precisely the counterterms required to cancelthe UV divergences in the gravitational effective action. In particular, theleading contribution to the entanglement entropy is given by the renormalizedBekenstein-Hawking formula, in accordance with a proposal of Susskind andUglum. We show that the subleading UV-divergent terms in the entanglemententropy depend nontrivially on the quantum state. We compute new subleadingterms in the entanglement entropy and find agreement with the Wald entropyformula for black hole spacetimes with bifurcate Killing horizons. We speculatethat the entanglement entropy of an arbitrary spatial boundary may be awell-defined observable in quantum gravity.

Renormalization of Entanglement Entropy and the Gravitational Effective Action