  We study quantum corrections to holographic entanglement entropy in
AdS$_3$/CFT$_2$; these are given by the bulk entanglement entropy across the
Ryu-Takayanagi surface for all fields in the effective gravitational theory. We
consider bulk $U(1)$ gauge fields and gravitons, whose dynamics in AdS$_3$ are
governed by Chern-Simons terms and are therefore topological. In this case the
relevant Hilbert space is that of the edge excitations. A novelty of the
holographic construction is that such modes live not only on the bulk
entanglement cut but also on the AdS boundary. We describe the interplay of
these excitations and provide an explicit map to the appropriate extended
Hilbert space. We compute the bulk entanglement entropy for the CFT vacuum
state and find that the effect of the bulk entanglement entropy is to
renormalize the relation between the effective holographic central charge and
Newton's constant. We also consider excited states obtained by acting with the
$U(1)$ current on the vacuum, and compute the difference in bulk entanglement
entropy between these states and the vacuum. We compute this UV-finite
difference both in the bulk and in the CFT finding a perfect agreement.
