  How do we uniquely identify a quantum phase, given its ground state
wave-function? This is a key question for many body theory especially when we
consider phases like topological insulators, that share the same symmetry but
differ at the level of topology. The entanglement spectrum has been proposed as
a ground state property that captures characteristic edge excitations. Here we
study the entanglement spectrum for topological band insulators. We first show
that insulators with topological surface states will necessarily also have
protected modes in the entanglement spectrum. Surprisingly, however, the
converse is not true. Protected entanglement modes can also appear for
insulators without physical surface states, in which case they capture a more
elusive property. This is illustrated by considering insulators with only
inversion symmetry. Inversion is shown to act in an unusual way, as an
antiunitary operator, on the entanglement spectrum, leading to this protection.
The entanglement degeneracies indicate a variety of different phases in
inversion symmetric insulators, and these phases are argued to be robust to the
introduction of interactions.
