  A definition for the entanglement entropy in a gauge theory was given
recently in arXiv:1501.02593. Working on a spatial lattice, it involves
embedding the physical state in an extended Hilbert space obtained by taking
the tensor product of the Hilbert space of states on each link of the lattice.
This extended Hilbert space admits a tensor product decomposition by definition
and allows a density matrix and entanglement entropy for the set of links of
interest to be defined. Here, we continue the study of this extended Hilbert
space definition with particular emphasis on the case of Non-Abelian gauge
theories.
  We extend the electric centre definition of Casini, Huerta and Rosabal to the
Non-Abelian case and find that it differs in an important term. We also find
that the entanglement entropy does not agree with the maximum number of Bell
pairs that can be extracted by the processes of entanglement distillation or
dilution, and give protocols which achieve the maximum bound. Finally, we
compute the topological entanglement entropy which follows from the extended
Hilbert space definition and show that it correctly reproduces the total
quantum dimension in a class of Toric code models based on Non-Abelian discrete
groups.
