  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.
