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Rényi entropies and negative central charges in non-Hermitian quantum systems
by Yi-Ting Tu, Yu-Chin Tzeng, Po-Yao Chang
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Submission summary
| Authors (as registered SciPost users): | Po-Yao Chang |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2107.13006v5 (pdf) |
| Date submitted: | 2022-05-03 04:45 |
| Submitted by: | Chang, Po-Yao |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
Quantum entanglement is one essential element to characterize many-body quantum systems. However, the entanglement measures are mostly discussed in Hermitian systems. Here, we propose a natural extension of entanglement and R\'enyi entropies to non-Hermitian quantum systems. There have been other proposals for the computation of these quantities, which are distinct from what is proposed in the current paper. We demonstrate the proposed entanglement quantities which are referred to as generic entanglement and R\'enyi entropies. These quantities capture the desired entanglement properties in non-Hermitian critical systems, where the low-energy properties are governed by the non-unitary conformal field theories (CFTs). We find excellent agreement between the numerical extrapolation of the negative central charges from the generic entanglement/R\'enyi entropy and the non-unitary CFT prediction. Furthermore, we apply the generic entanglement/R\'enyi entropy to symmetry-protected topological phases with non-Hermitian perturbations. We find the generic $n$-th R\'enyi entropy captures the expected entanglement property, whereas the traditional R\'enyi entropy can exhibit unnatural singularities due to its improper definition.
Author comments upon resubmission
We thank the two referees for their careful reading of the manuscript and for their comments.
Reply to Referee A
Regarding the comments 1-3, we thank the referee for pointing out our inconsistency of discussing the negative entanglement entropy. We rewrite the introduction part and point out the possibility to have negative entanglement entropy is from the new definition of the density matrix involving the left and right eigenvectors. We also emphasize that Refs. [20-22] are the first few papers to obtained the negative entanglement/Renyi entropy and our new proposed entanglement measures are in complementary to these existing approaches.
Reply to the requested changes
1) We mention the previous proposal Refs. [20-22] regarding to their methods the results in the revised manuscript.
2) The interpretation of the “negative” entropy is already discussed in the conclusion and discussion. We point out the non-Hermiticity can be introduced by post-selection and the post-selected entropy can be negative.
3) We thank the referee for the suggestion. We add the sentence in the abstract that the referee suggested.
4) We thank the referee for the suggestion. We follow the referee’s suggestion.
5) We add the footnote to clarify this point.
6) Yes, the eigenvalues of the reduced density matrix are real or conjugate pairs, which lead to the real outcomes. We add this sentence in the revised manuscript.
7) We thank the referee for pointing out this reference. We will investigate the complex periodic Ising chain in detail for future paper.
Reply to Referee B
We thank the Referee’s suggestion. We correct our inaccuracy of the review and revised the manuscript accordingly. We thank the referee for pointing out the expression of our generic Renyi entropy can have the geometric interpretation in the n-sheeted Riemann surface with partial-time reversal transformation.
Reply to the requested changes:
- We rewrite the introduction part according to the Referee’s suggestion.
- We add detailed discussion about the central charge in Section 2.2.
List of changes
1. We correct our misinterpretation of the literature reviews in the abstract and introduction parts. We extend the introduction part regarding to the existing proposals (Refs.~[17-21]).
2. In section 2, we add one sentence discussing why the outcomes of the generic entanglement/Renyi entropy are real.
3. We add detail discussion regarding the the section charge in Sec. 2.2 and add one reference.
V. Pasquier and H. Saleur, Common structures between finite systems and confor- mal field theories through quantum groups, Nuclear Physics B 330(2), 523 (1990), doi:https://doi.org/10.1016/0550-3213(90)90122-T.
4. We add one new expression of the generic Renyi entropy (Eq. (5)) as pointing out by the second Referee.
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 4) on 2022-5-11 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2107.13006v5, delivered 2022-05-11, doi: 10.21468/SciPost.Report.5065
Report
In this resubmission, the Introduction now correctly describes the literature on entanglement measures for non-Hermitian systems. However, the discussion of the central charge in Section 2.2 still suffers from the confusion I pointed out in my previous report -- defining a 1+1d model only by its quantum Hamiltonian does not determine the central charge, because this only gives $L_0+\bar{L}_0$, and not the full Virasoro algebra. Hence, the statement that "the Hamiltonian is the usual XXZ chain with central charge c = 1" makes no sense. Moreover, the central charge is a local bulk quantity, and hence it does not depend on boundary conditions.
To solve this confusion, I can suggest to use the following statements :
- The Hamiltonian (3) [or its periodic variant] is the anisotropic limit of an integrable six-vertex (6V) model with complex Boltzmann weights, with central charge $c=1-\frac{6\theta^2}{\pi(\pi-\theta)}$.
- In this complex-weight 6V model, the phase factors cancel everywhere, except at boundaries and along lines connecting conical singularities.
- The trace operation consistent with the complex-weight 6V model is the modified trace, which includes a factor $q^{-2\sigma_A^z}$.
- Alternatively, the Hamiltonian (3) can be viewed as the anisotropic limit of a 6V model with real Boltzmann weights, with central charge c=1, and non-trivial complex boundary conditions. The corresponding trace operation is the usual trace.
Requested changes
Correct the discussion on the central charge in Section 2.2.
Report #1 by Anonymous (Referee 3) on 2022-5-6 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2107.13006v5, delivered 2022-05-06, doi: 10.21468/SciPost.Report.5035
Strengths
Same as in my original report.
Weaknesses
These have now been addressed by introducing all changes I had requested in my original report.
Report
As I wrote in my original report, I think that the paper deserves publication in SciPost as it makes a novel and positive contribution to the development of entanglement measures that are appropriate for non-unitary systems.
The paper is well written and contains many relevant examples.
My main criticisms related to how the authors discussed their work in relation to previous contributions in the area. My general feeling was that some of their writing seemed unduly negative about previous proposals and that those were not always properly acknowledged.
I am happy the authors have considered my comments and acted on them. Consequently, I am happy with the paper in its current form and recommend publication without further changes.