Entanglement Entropy in Pure $Z_2$ Gauge Lattices

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.