  Entanglement entropy, taken here to be geometric, requires a geometrically
separable Hilbert space. In lattice gauge theories, it is not immediately clear
if the physical Hilbert space is geometrically separable. In a previous paper
we have shown that the physical Hilbert space in pure gauge abelian lattice
theories exhibits some form of geometric scaling with the lattice volume, which
suggest that the space is locally factorizable and, therefore, geometrically
separable. In this paper, we provide strong evidence that indicates that this
scaling is not present when the group is non-abelian. We do so by looking at
the scaling of the dimension of the physical Hilbert space of theories with
certain discrete groups. The lack of an appropriate scaling implies that the
physical Hilbert space of such a theory does not admit a local factorization.
We then extend the reasoning, as sensibly possible, to SU(2) and SU(N) to reach
the same conclusion. Lastly, we show that the addition of matter fields to
non-abelian lattice gauge theories makes the resulting physical Hilbert space
locally factorizable.
