# Entanglement, Non-Hermiticity, and Duality

### Submission summary

 As Contributors: Peng Ye Preprint link: scipost_202102_00006v2 Date accepted: 2021-06-24 Date submitted: 2021-06-11 17:06 Submitted by: Ye, Peng Submitted to: SciPost Physics Academic field: Physics Specialties: Condensed Matter Physics - Theory Approach: Theoretical

### Abstract

Usually duality process keeps energy spectrum invariant. In this paper, we provide a duality, which keeps entanglement spectrum invariant, in order to diagnose quantum entanglement of non-Hermitian non-interacting fermionic systems. We limit our attention to non-Hermitian systems with a complete set of biorthonormal eigenvectors and an entirely real energy spectrum. The original system has a reduced density matrix $\rho_\mathrm{o}$ and the real space is partitioned via a projecting operator $\mathcal{R}_{\mathrm o}$. After dualization, we obtain a new reduced density matrix $\rho_{\mathrm{d}}$ and a new real space projector $\mathcal{R}_{\mathrm d}$. Remarkably, entanglement spectrum and entanglement entropy keep invariant. Inspired by the duality, we defined two types of non-Hermitian models, upon $\mathcal R_{\mathrm{o}}$ is given. In type-I exemplified by the non-reciprocal model'', there exists at least one duality such that $\rho_{\mathrm{d}}$ is Hermitian. In other words, entanglement information of type-I non-Hermitian models with a given $\mathcal{R}_{\mathrm{o}}$ is entirely controlled by Hermitian models with $\mathcal{R}_{\mathrm{d}}$. As a result, we are allowed to apply known results of Hermitian systems to efficiently obtain entanglement properties of type-I models. On the other hand, the duals of type-II models, exemplified by non-Hermitian Su-Schrieffer-Heeger model'', are always non-Hermitian. For the practical purpose, the duality provides a potentially \textit{efficient} computation route to entanglement of non-Hermitian systems. Via connecting different models, the duality also sheds lights on either trivial or nontrivial role of non-Hermiticity played in quantum entanglement, paving the way to potentially systematic classification and characterization of non-Hermitian systems from the entanglement perspective.

Published as SciPost Phys. 11, 003 (2021)

Dear Editor:
Hereby we are resubmitting our work to SciPost Physics.

We thank you for sending us two referee reports. We have carefully read the reports and provided detailed replies, and especially point-to-point response. We thank two referees for their positive comments on scientific merit and creativity of our work. We made careful revision by following all constructive suggestions in the two reports.

Thanks a lot!

Bests

### List of changes

We list main changes below.
1. Reference section was updated. The papers suggested by referees have been taken care of.
2. Abstract section was rewritten.
3. A useful table (Table 1) was added.
4. At the end of Section-4, a paragraph was added for explaining multiple issues.
5. Several typos were identified and corrected.

Description of changes is also available in our replies. Some important changes were marked in red in the revised draft.

Thanks!

### Submission & Refereeing History

Resubmission scipost_202102_00006v2 on 11 June 2021
Submission scipost_202102_00006v1 on 3 February 2021

## Reports on this Submission

### Report

The authors have addressed my comments in detail. I find their response persuasive and think that their results are important and sound on the entanglement of certain non-hermitian systems. Therefore, I'd recommend publication in Scipost Physics.

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

### Strengths

Unchanged from first report

### Weaknesses

No mayor weaknesses as all points I raised in my first report have been addressed.

### Report

I am satisfied with the changes made by the authors as they have addressed all my queries in detail and improved the paper in several ways, including by expanding their set of references .

I am happy to accept the paper in its current form.

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