Entanglement in a pure state of a many-body system can be characterized by the Renyi entropies $S^{(alpha)}=lntextrm{tr}(rho^alpha)/(1-alpha)$ of the reduced density matrix $rho$ 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 Renyi 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.
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