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Bounds on the entanglement entropy by the number entropy in non-interacting fermionic systems

Entanglement in a pure state of a many-body system can be characterized by the Rényi entropies $S^{(α)}=\ln\textrm{tr}(ρ^α)/(1-α)$ of the reduced density matrix $ρ$ of a subsystem. These entropies are, however, difficult to access experimentally and can typically be determined for small systems only. Here we show that for free fermionic systems in a Gaussian state and with particle number conservation, $\ln S^{(2)}$ can be tightly bound by the much easier accessible Rényi number entropy $S^{(2)}_N=-\ln \sum_n p^2(n)$ which is a function of the probability distribution $p(n)$ of the total particle number in the considered subsystem only. A dynamical growth in entanglement, in particular, is therefore always accompanied by a growth---albeit logarithmically slower---of the number entropy. We illustrate this relation by presenting numerical results for quenches in non-interacting one-dimensional lattice models including disorder-free, Anderson-localized, and critical systems with off-diagonal disorder.