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Action and entanglement in gravity and field theory

In non-gravitational quantum field theory, the entanglement entropy across a surface depends on the short-distance regularization. Quantum gravity should not require such regularization, and it's been conjectured that the entanglement entropy there is always given by the black hole entropy formula evaluated on the entangling surface. We show that these statements have precise classical counterparts at the level of the action. Specifically, we point out that the action can have a non-additive imaginary part. In gravity, the latter is fixed by the black hole entropy formula, while in non-gravitating theories, it is arbitrary. From these classical facts, the entanglement entropy conjecture follows by heuristically applying the relation between actions and wavefunctions.