  We study the impurity entanglement entropy $S_e$ in quantum impurity models
that feature a Kondo-destruction quantum critical point (QCP) arising from a
pseudogap in the conduction-band density of states or from coupling to a
bosonic bath. On the local-moment (Kondo-destroyed) side of the QCP, the
entanglement entropy contains a critical component that can be related to the
order parameter characterizing the quantum phase transition. In Kondo models
describing a spin-$\Simp$, $S_e$ assumes its maximal value of $\ln(2\Simp+1)$
at the QCP and throughout the Kondo phase, independent of features such as
particle-hole symmetry and under- or over-screening. In Anderson models, $S_e$
is nonuniversal at the QCP, and at particle-hole symmetry, rises monotonically
on passage from the local-moment phase to the Kondo phase; breaking this
symmetry can lead to a cusp peak in $S_e$ due to a divergent charge
susceptibility at the QCP. Implications of these results for quantum critical
systems and quantum dots are discussed.
