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

### Submission summary

 As Contributors: Michael Fleischhauer · Maximilian Kiefer-Emmanouilidis · Jesko Sirker Preprint link: scipost_202003_00048v2 Date accepted: 2020-05-13 Date submitted: 2020-04-28 02:00 Submitted by: Sirker, Jesko Submitted to: SciPost Physics Academic field: Physics Specialties: Atomic, Molecular and Optical Physics - Theory Condensed Matter Physics - Theory Quantum Physics Approach: Theoretical

### Abstract

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.

### Ontology / Topics

See full Ontology or Topics database.

Published as SciPost Phys. 8, 083 (2020)

### Author comments upon resubmission

First of all, we would like to thank both referees for the very
careful and thorough evaluation of our work and the helpful
comments. In the following we will provide a detailed and point by
point reply. We have modified our manuscript accordingly and followed
most of the suggestions made. The few cases where we deviated from the
recommendations are explained as well.

### List of changes

List of changes:

1) Abstract: "tightly bound---from above and below---by the"

2) Refs. [2-6] and discussion of experimental techniques to measure entanglement added.

3) Below Eq. (1): Definition of p(n) added.

4) Refs. [12-14] added.

5) Discussion of cumulants added at the bottom of page 2. Refs. [18,19] added.

7) Discussion of consequences for the interacting case added at the top of page 3. Ref. [21] added.

8) Below Eq. (9): p(n) defined as Fourier coefficients of the moment generating function.

9) Sec. 3.2 and caption of Fig. 2: Explanations about the disorder average added.

10) Sec 3.3.: Clarified that the amplitudes are drawn from a box distribution.

11) Sec 3.3.: Critical exponent Psi=1/2 specified.

12) Sec. 3.3.: System size added, clarified that results are converged in the time window shown.

13) Conclusions: Discussion about the relevance of the results for systems with interactions and disorder added.

### Submission & Refereeing History

You are currently on this page

Resubmission scipost_202003_00048v2 on 28 April 2020
Submission scipost_202003_00048v1 on 16 March 2020

## Reports on this Submission

### Strengths

As in my previous report

### Weaknesses

Weaknesses pointed out in previous reports have been addressed

### Report

In my opinion the Authors have addressed well all points previously made by both Referees. Considering my previous assessment of the work, I would now recommend its publication.

### Requested changes

None

• validity: high
• significance: high
• originality: good
• clarity: good
• formatting: excellent
• grammar: excellent

### Report

The authors replied to all the points raised in a satisfactory and exhaustive way, and I believe the paper deserves publication in Scipost Physics.

• validity: high
• significance: high
• originality: good
• clarity: high
• formatting: excellent
• grammar: excellent