Exact relations between particle fluctuations and entanglement in Fermi gases

We derive exact relations between the Renyi entanglement entropies and the particle-number fluctuations of (connected and disjoint) spatial regions in systems of N non-interacting fermions in arbitrary dimension. We prove that the asymptotic large-N behavior of the entanglement entropies is proportional to the variance of the particle number. We also consider 1D Fermi gases with a localized impurity, where all particle cumulants contribute to the asymptotic large-N behavior of the entanglement entropies. The particle cumulant expansion turns out to be convergent for all integer-order Renyi entropies (except for the von Neumann entropy) and the first few cumulants provide already a good approximation. Since the particle cumulants are accessible to experiments, these relations may provide a measure of entanglement in these systems. Copyright (C) EPLA, 2012