The scaling of entanglement entropy, and more recently the full entanglement spectrum, have become useful tools for characterizing certain universal features of quantum many-body systems.
Although entanglement entropy is difficult to measure experimentally, we show that for systems that can be mapped to non-interacting fermions both the von Neumann entanglement entropy and generalized Renyi entropies can be related exactly to the cumulants of number fluctuations, which are accessible experimentally. Such systems include free fermions in all dimensions, the integer quantum Hall states and topological insulators in two dimensions, strongly repulsive bosons in one-dimensional optical lattices, and the spin-1/2 XX chain, both pure and strongly disordered.
The same formalism can be used for analyzing entanglement entropy generation in quantum point contacts with non-interacting electron reservoirs. Beyond the non-interacting case, we show that the scaling of fluctuations in one-dimensional critical systems behaves quite similarly to the entanglement entropy, and in analogy to the full counting statistics used in mesoscopic transport, give important information about the system. The behavior of fluctuations, which are the essential feature of quantum systems, are explained in a general framework and analyzed in a variety of specific situations.