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

• Main Author: Maximilian Kiefer-Emmanouilidis, Razmik Unanyan, Jesko Sirker, Michael Fleischhauer
• Format: Article
• Language: English
• Published: SciPost 2020-06-01
• Series: SciPost Physics
• Online Access: https://scipost.org/SciPostPhys.8.6.083

Entanglement in a pure state of a many-body system can be characterized by the R\'enyi entropies $S^{(\alpha)}=\ln\textrm{tr}(\rho^\alpha)/(1-\alpha)$ 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---from above and below---by the much easier accessible R\'enyi 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.