R\'{e}nyi Entanglement Entropies and the Entanglement Spectrum

We describe a simple method for computing the full entanglement spectrum of any finite density matrix from the R\'{e}nyi entropies of integer order. This has important implications for non-interacting fermionic systems where the R\'{e}nyi entropies are directly related to the cumulants of charge number fluctuations, and for quantum Monte Carlo simulations where it is now becoming possible to compute the R\'{e}nyi entropies but not the von Neumann entropy or the full entanglement spectrum.