  We focus on the issue of proper definition of entanglement entropy in lattice
gauge theories, and examine a naive definition where gauge invariant states are
viewed as elements of an extended Hilbert space which contains gauge
non-invariant states as well. Working in the extended Hilbert space, we can
define entanglement entropy associated with an arbitrary subset of links, not
only for abelian but also for non-abelian theories. We then derive the
associated replica formula. We also discuss the issue of gauge invariance of
the entanglement entropy. In the $Z_N$ gauge theories in arbitrary space
dimensions, we show that all the standard properties of the entanglement
entropy, e.g. the strong subadditivity, hold in our definition. We study the
entanglement entropy for special states, including the topological states for
the $Z_N$ gauge theories in arbitrary dimensions. We discuss relations of our
definition to other proposals.
