Valence Bond and von Neumann Entanglement Entropy in Heisenberg Ladders

We present a direct comparison of the recently-proposed valence bond entanglement entropy and the von Neumann entanglement entropy on spin 1/2 Heisenberg systems using quantum Monte Carlo and density-matrix renormalization group simulations. For one-dimensional chains we show that the valence bond entropy can be either less or greater than the von Neumann entropy, hence it cannot provide a bound on the latter. On ladder geometries, simulations with up to seven legs are sufficient to indicate that the von Neumann entropy in two dimensions obeys an area law, even though the valence bond entanglement entropy has a multiplicative logarithmic correction.