  We show that the Hilbert space of physical states on a pure $Z_2$ gauge
lattice in $1 + 1$ and $2 + 1$ dimensions is geometrically separable if the
fundamental physical degrees of freedom are taken to be the plaquettes. This
results in a physical entanglement entropy that is not affected by gauge
fixing. We introduce a lattice model that is physically equivalent to the
original and whose entanglement entropy, calculated using link degrees of
freedom, is the same as the entanglement entropy calculated using physical
states. We also show that, for non-physical gauge link states, entanglement
entropy quantifies constraints between gauge choices in plaquettes adjacent to
the boundary.
