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Local Operator Entanglement in Spin Chains

by Eric Mascot, Masahiro Nozaki, Masaki Tezuka

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Submission summary

Authors (as registered SciPost users): Eric Mascot
Submission information
Preprint Link:  (pdf)
Date accepted: 2023-09-22
Date submitted: 2023-08-28 04:03
Submitted by: Mascot, Eric
Submitted to: SciPost Physics Core
Ontological classification
Academic field: Physics
  • Information Theory
  • Condensed Matter Physics - Theory
  • High-Energy Physics - Theory
  • Quantum Physics
Approach: Theoretical


Understanding how and whether local perturbations can affect the entire quantum system is a fundamental step in understanding non-equilibrium phenomena such as thermalization. This knowledge of non-equilibrium phenomena is applicable for quantum computation, as many quantum computers employ non-equilibrium processes for computations. In this paper, we investigate the evolution of bi- and tripartite operator mutual information of the time-evolution operator and the Pauli spin operators in the one-dimensional Ising model with magnetic field and the disordered Heisenberg model to study the properties of quantum circuits. In the Ising model, the early-time evolution qualitatively follows an effective light cone picture, and the late-time value is well described by Page's value for a random pure state. In the Heisenberg model with strong disorder, we find that many-body localization prevents the information from propagating and being delocalized. We also find an effective Ising Hamiltonian that describes the time evolution of bi- and tripartite operator mutual information for the Heisenberg model in the large disorder regime.

Author comments upon resubmission

Dear editor-in-charge,

I am writing to resubmit the revised version of our manuscript titled "Local Operator Entanglement in Spin Chains”. We are grateful for the thoughtful feedback provided by the referees, and we believe that the revisions we have made address the issues raised comprehensively.

We have carefully revised the manuscript according to the reviewer's feedback, and we have also made additional improvements to enhance the clarity, analysis, and overall contribution of the work. Specifically, we have made the following changes:

  1. We have restructured the manuscript to focus on the “full overlap” configuration and moved other configurations to the appendix.

  2. We have included additional analysis and discussion.

  3. We have carefully proofread the manuscript to eliminate any grammatical errors or typographical mistakes.

We believe that these revisions align the manuscript much more closely with the scope and standards of SciPost Physics Core, and we are confident that the updated version represents a substantial contribution to the field. We appreciate the opportunity to resubmit our work to SciPost Physics Core and are hopeful for a positive outcome.

Sincerely, Eric Mascot Masahiro Nozaki Masaki Tezuka

List of changes

- Revised the first few sentences of the abstract
- Revised the introduction to provide clear motivation and to better articulate the results
- Provided additional analysis of the data
- Improved the description of thermalization
- Improved the description of OTOC
- Clarified "robust against noise" for the MBL phase
- Corrected operators with maximal entanglement structure
- Added a discussion of EPR pairs in a subsystem as an interpretation of BOMI and TOMI
- Added a discussion of factorization of the density matrix as an interpretation of late-time BOMI
- Moved "partial overlap" and "disjoint" configurations to the appendix
- Added a discussion on the effective model for large disorder
- Many minor corrections

Published as SciPost Phys. Core 6, 070 (2023)

Reports on this Submission

Anonymous Report 1 on 2023-9-7 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2012.14609v5, delivered 2023-09-07, doi: 10.21468/SciPost.Report.7778


The authors have made efforts to address the comments. The essential structure of the paper remains the same and still feels a little unbalanced, with more qualitative discussion and general review (e.g. of MBL) than may be necessary (although the authors have made some effort to lighten the manuscript by taking some of the results out of the main text). It remains true that many of the take-aways from the numerical results are rather qualitative, and it is not clear to me that they go beyond previous theoretical analysis such as the reference by Kudler-Flam et al. Nevertheless the numerical results may be a reference point for future works. The background material may also be useful for some readers. On these bases the paper may now be publishable in Core.

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Anonymous on 2023-09-08  [id 3964]

The paper investigates the operator entanglement of the time-evolution operator for chaotic, integrable and many-body localized systems.  It focuses on bipartite and tripartite operator mutual information to quantify the entangling properties of the time evolution. By mapping the density matrix to a channel and using that to numerically evaluate the mutual information, the authors are able to describe some of the universal features of the dynamics. For this they consider the clean spin-1/2 Ising model in its integrable and non-integrable limits, and the random-field Heisenberg model, for studying many-body localization (MBL). They have investigated the early and late behaviour of these observables in the different dynamical regimes. 

Their findings suggest that in the clean case that the short time behaviour is consistent with a light cone and at late times the state fits the Page curve. They draw on similarities with 2D holographic CFTs to explain their results. While this would be reasonable, but the field theory may not apply to spin chain in question. There is a whole body of literature studying chaos and light cone dynamics in spin chains. An attempt should be made to connect the results with the known results in the literature.  In the integrable limit, the quantities do not decay with time. Can this be understood analytically even from the perspective of free-field theory? The connection to free field theory can be made more thorough. In this case as well, there is a large body of work studying entanglement dynamics in integrable spin chains, and connecting their results with the literature would be quite appropriate. 

In the disordered model, at weak disorder the results are consistent with chaotic dynamics. While at strong disorder in the localized phase the oscillations can be described by a simple effective Ising model. The number of disorder realizations (20) is too low for this analysis and therefore the error bars are large. It would be reasonable to average over more disorder realizations. 

In both the cases (clean and disordered) some scaling with system (L=12, 14) can be attempted. It should not be computationally prohibitively expensive. 

Overall, the manuscript has some new results which provide additional understanding of chaotic and many-body localized systems in terms of operator entanglement. There are three main points to address 1. Discussion relating to known results on entanglement dynamics in chaotic and integrable spin chains. 2. Finite size scaling of the results. 3. Reduce error bars by using a larger number of disorder realizations.

With these changes I can recommend the manuscript for publication.