  Despite the seeming simplicity of the theory, calculating (and even defining)
entanglement entropy for the Maxwell theory of a $U(1)$ gauge field in (3+1)
dimensions has been the subject of controversy. It is generally accepted that
the ground state entanglement entropy for a region of linear size $L$ behaves
as an area law with a subleading logarithm, $S = \alpha L^2 -\gamma \log L$.
While the logarithmic coefficient $\gamma$ is believed to be universal, there
has been disagreement about its precise value. After carefully accounting for
subtle boundary corrections, multiple analyses in the high energy literature
have converged on an answer related to the conformal trace anomaly, which is
only sensitive to the local curvature of the partition. In contrast, a
condensed matter treatment of the problem yielded a topological contribution
which is not captured by the conformal field theory calculation. In this
perspective piece, we review aspects of the various calculations and discuss
the resolution of the discrepancy, emphasizing the important role played by
charged states (the "extended Hilbert space") in defining entanglement for a
gauge theory. While the trace anomaly result is sufficient for a strictly pure
gauge field, coupling the gauge field to dynamical charges of mass $m$ gives a
topological contribution to $\gamma$ which survives even in the
$m\rightarrow\infty$ limit. For many situations, the topological contribution
from dynamical charges is physically meaningful and should be taken into
account. We also comment on other common issues of entanglement in gauge
theories, such as entanglement distillation, algebraic definitions of
entanglement, and gauge-fixing procedures.
