  We explore the question of finiteness of the entanglement entropy in
gravitational theories whose emergent space is the target space of a
holographic dual. In the well studied duality of two-dimensional non-critical
string theory and $c=1$ matrix model, this question has been studied earlier
using fermionic many-body theory in the space of eigenvalues. The entanglement
entropy of a subregion of the eigenvalue space, which is the target space
entanglement in the matrix model, is finite, with the scale being provided by
the local Fermi momentum. The Fermi momentum is, however, a position-dependent
string coupling, as is clear in the collective field theory formulation. This
suggests that the finiteness is a non-perturbative effect. We provide evidence
for this expectation by an explicit calculation in the collective field theory
of matrix quantum mechanics with vanishing potential. The leading term in the
cumulant expansion of the entanglement entropy is calculated using exact
eigenstates and eigenvalues of the collective Hamiltonian, yielding a finite
result, in precise agreement with the fermion answer. Treating the theory
perturbatively, we show that each term in the perturbation expansion is UV
divergent. However the series can be resummed, yielding the exact finite
result. Our results indicate that the finiteness of the entanglement entropy
for higher dimensional string theories is non-perturbative as well, with the
scale provided by the Newton constant.
