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

Abstract

Entanglement in a pure state of a many-body system can be characterized by the Rényi entropies $S^{(\alpha )}=\ln \text{tr}({\rho }^{\alpha })/(1-\alpha )$ 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 Rényi number entropy $S_N^{(2)}=-\ln {\sum }_n{p}^2(n)$ 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.