  A definition for the entanglement entropy in both Abelian and non-Abelian
gauge theories has been given in the literature, based on an extended Hilbert
space construction. The result can be expressed as a sum of two terms, a
classical term and a quantum term. It has been argued that only the quantum
term is extractable through the processes of quantum distillation and dilution.
Here we consider gauge theories in the continuum limit and argue that quite
generically, the classical piece is dominated by modes with very high momentum,
of order the cut-off, in the direction normal to the entangling surface. As a
result, we find that the classical term does not contribute to the relative
entropy or the mutual information, in the continuum limit, for states which
only carry a finite amount of energy above the ground state. We extend these
considerations for $p$-form theories, and also discuss some aspects pertaining
to electric-magnetic duality.

High Energy Physics - Theory;Statistical Mechanics;High Energy Physics - Lattice;Quantum Physics