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(α) = ln tr(ρα)/(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, S(2) can be tightly bound—from above and below—by the much easier accessible Rényi number entropy S(N2) = − ln Pn p2(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 noninteracting one-dimensional lattice models including disorder-free, Anderson-localized, and critical systems with off-diagonal (bond) disorder.