Improving entanglement and thermodynamic R\'enyi entropy measurements in quantum Monte Carlo

We present a method for improving measurements of the entanglement R\'enyi entropies in quantum Monte Carlo simulations by relating them with measurements of participation R\'enyi entropies. Exploiting the capability of building improved estimators for the latter allows to obtain very good estimates for entanglement R\'enyi entropies. When considering a full system instead of a bipartition, the method can be further ameliorated providing access to the thermodynamic R\'enyi entropies with high accuracy. We also explore a recently-proposed method for the reconstruction of the entanglement spectrum from entanglement R\'enyi entropies and finally show how potential entanglement Hamiltonians may be tested for their validity using a comparison with thermal R\'enyi entropies.