  The valence-bond structure of spin-1/2 Heisenberg antiferromagnets is closely
related to quantum entanglement. We investigate measures of entanglement
entropy based on transition graphs, which characterize state overlaps in the
overcomplete valence-bond basis. The transition graphs can be generated using
projector Monte Carlo simulations of ground states of specific hamiltonians or
using importance-sampling of valence-bond configurations of amplitude-product
states. We consider definitions of entanglement entropy based on the bonds or
loops shared by two subsystems (bipartite entanglement). Results for the
bond-based definition agrees with a previously studied definition using
valence-bond wave functions (instead of the transition graphs, which involve
two states). For the one dimensional Heisenberg chain, with uniform or random
coupling constants, the prefactor of the logarithmic divergence with the size
of the smaller subsystem agrees with exact results. For the ground state of the
two-dimensional Heisenberg model (and also Neel-ordered amplitude-product
states), there is a similar multiplicative violation of the area law. In
contrast, the loop-based entropy obeys the area law in two dimensions, while
still violating it in one dimension - both behaviors in accord with
expectations for proper measures of entanglement entropy.
