  We study the entanglement structure of lattice gauge theories from the local
operational point of view, and, similar to Soni and Trivedi (arXiv:1510.07455),
we show that the usual entanglement entropy for a spatial bipartition can be
written as the sum of an undistillable gauge part and of another part
corresponding to the LOCC distillable entanglement, which is obtained by
depolarizing the local superselection sectors. We demonstrate that the
distillable entanglement is zero for pure abelian gauge theories in the weak
coupling limit, while it is in general nonzero for the nonabelian case. We also
consider gauge theories with matter, and show in a perturbative approach how
area laws -- including a topological correction -- emerge for the distillable
entanglement.
