  We study the entanglement entropy and entanglement spectrum of the
paradigmatic Bose-Hubbard model, describing strongly correlated bosons on a
lattice. The use of a controlled approximation - the slave-boson approach -
allows us to study entanglement in all regimes of the model (and, most
importantly, across its superfluid/Mott-insulator transition) at a minimal
cost. We find that the area-law scaling of entanglement -- verified in all the
phases -- exhibits a sharp singularity at the transition. The singularity is
greatly enhanced when the transition is crossed at fixed, integer filling, due
to a richer entanglement spectrum containing an additional gapless mode, which
descends from the amplitude (Higgs) mode of the global excitation spectrum --
while this mode remains gapped at the generic (commensurate-incommensurate)
transition with variable filling. Hence the entanglement properties contain a
unique signature of the two different forms of bosonic criticality exhibited by
the Bose-Hubbard model.
