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Local Operator Entanglement in Spin Chains
by Eric Mascot, Masahiro Nozaki, Masaki Tezuka
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Authors (as registered SciPost users):  Eric Mascot 
Submission information  

Preprint Link:  https://arxiv.org/abs/2012.14609v3 (pdf) 
Date submitted:  20210120 08:02 
Submitted by:  Mascot, Eric 
Submitted to:  SciPost Physics 
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Academic field:  Physics 
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Approach:  Theoretical 
Abstract
We study the time evolution of bi and tripartite operator mutual information of the timeevolution operator and Pauli's spin operators in the onedimensional Ising model with magnetic field and the disordered Heisenberg model. In the Ising model, the earlytime evolution qualitatively follows an effective light cone picture, and the latetime value is well described by Page's value for a random pure state. In the Heisenberg model with strong disorder, we find manybody localization prevents the information from propagating and being delocalized. We also find an effective Ising Hamiltonian describes the time evolution of bi and tripartite operator mutual information for the Heisenberg model in the large disorder regime.
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Reports on this Submission
Anonymous Report 2 on 2021512 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2012.14609v3, delivered 20210511, doi: 10.21468/SciPost.Report.2905
Report
The authors study the dynamics of bipartite and tripartite operator entanglement entropy of one dimensional spin1/2 models. These measures have been studied for clean chaotic and integrable cases, along with chaotic and MBL phases of disordered spin chains. These multipartite measures can provide a more granular understanding of these wellknown phases of matter. The main conclusion of the work is characterising the steady state in these cases, in particular finding evidence for the Page curve of a random pure state in the clean chaotic regime.
Although the topic is of interest in the context of operators spreading, in my opinion the current version of the manuscript hasn't made significant progress to warrant publication in its current form. It is unclear in what sense the results go beyond the existing knowledge about these phenomena. It is likely that some new insights can be gleaned from the numerical data, but in the absence of significant theoretical analysis the impact of the numerics on its own is inconclusive.
I suggest a few theoretical questions which could sharpen the results of the paper.
1. Do we expect change in behaviour in TOMI in nonintegrable and integrable cases? If yes, why?
2. Does the Page curve apply also in the integrable and MBL cases? The steady states have volume law entanglement in both cases? How should we understand this theoretically?
3. MBL is associated with logarithmic growth of von Neumann entanglement entropy? How does that effect the results on BOMI and TOMI?
As a practical point, the error bars for the data in the disordered case are quite large. In its current form it is hard to draw any meaningful conclusion from the data. I expect with more disorder averaging the trends will be much clearer.
In its present form, the paper includes a lot of data and plots but not as much physical interpretation. It may even be feasible to reduce the number of figures in order to provide a sharper focus on the main physical consequences of the numerical results.
Anonymous Report 1 on 2021430 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2012.14609v3, delivered 20210429, doi: 10.21468/SciPost.Report.2857
Report
This paper presents numerical data for entanglement measures in a chaotic and a disordered chain, namely the mutual information and the “tripartite mutual information” for timeevolving operators. It is clear that considerable work has gone into the generation of the data and the presentation of this data and of background material. However, this manuscript does not appear to me to be publishable in Scipost. What is lacking is (a) a clearly defined motivation for studying these quantities and (b) clear lessons from the numerical data. The reader is shown many plots without necessarily knowing what to infer from them.
The authors do make attempts to interpret some of the findings (they describe a “light cone picture”, they use some basic properties of strongly entangled states, and they invoke the lbit picture for manybody localization in the disordered case). However these theoretical accounts seem relatively limited. For example the discussion of the chaotic phase is not quantitative (except in the relatively simple t>∞ limit).
The observables considered are linear combinations of operator entanglement entropies. Therefore in the chaotic phase existing ideas can be used to give a more quantitative account of their behavior, to the extent that this is universal at large scales. The scaling theory for operator entanglement in chaotic chains of Ref 19 applies here.
Regardless of the presence or absence of a theoretical explanation, clearer physical motivation for studying these quantities is needed. If I understand correctly, the idea behind the tripartite information of a channel in previous work was to determine whether information that is initially encoded locally is encoded locally or globally after the application of the channel. In the present paper the context is different (local operators) and the motivation is less clear to me. It is true that the local operators studied here happen to be unitary, so they can be viewed as channels if desired, but the authors do not give an argument that it is useful to do this.
The paper is currently rather long. The authors might wish to decide what the key message(s) of their paper should be, and cut down the length and the number of figures in the main text to give the paper some focus.
Here are more specific comments.
Eq 1  I understand the general idea the authors are trying to convey, but this equation is not correct and neither is the claimed relation between energy and temperature, in general.
Eq 3  missing normalization
Eq 5  can this be reformulated to be clearer?
Eq 9  this quantity is claimed to be “exponentially large”  isn’t it bounded and of order 1 in the models studied here?
Sec 2.2. “we expect quantum information to be preserved locally”. Generically only classical information is preserved locally in the MBL phase (the zcomponent of the lbit is conserved but not the xcomponent).
p6 “robust against noise” What is meant here? Noise (timedependent fluctuations in the Hamiltonian) will generically destroy the MBL phase.
“linear in noninteracting systems or general nonintegrable systems without localization”. This statement assumes translational invariance. Otherwise, sublinear growth is possible, even without localization.
p9 “unless O is the identity operator” This is not correct, there are other O with this property
p10, 11 contain many vague (e.g. “unrelated to each other”, “data regarding the local entanglement structure”) or repetitive statements
p19 should S_A,B be I_A,B?
Author: Eric Mascot on 20230825 [id 3926]
(in reply to Report 1 on 20210430)
We would like to express our sincere gratitude for taking the time to read our paper and for providing us with your invaluable feedback. We greatly appreciate your numerous insightful comments. We have addressed the key points, (a) providing a clear motivation, and (b) enhancing the clarity and comprehensibility of our analysis of the numerical data. To address the first point, we have incorporated an additional paragraph at the end of the introduction, which explicitly outlines the motivation behind our investigation of BOMI and TOMI. Regarding the second point, we have reworked the summary on page 3 to better articulate the results obtained in our paper. Furthermore, we have diligently reviewed your suggestions and made substantial revisions throughout the entire paper. Here, we briefly explain the motivation and analysis.
・Motivation for the Ising chain with magnetic field: We would like to study whether the late time evolution of the BOMI and TOMI of local operators depends on the operators in systems with the strong scrambling ability and few degrees of freedom. In twodimensional holographic CFT, systems with the strong scrambling effect and large degrees of freedom have a late time value of BOMI and TOMI that does not depend on the local operator. Since the initial state considered in this paper is characterized by the Pauli operators, the fact that the late time BOMI and TOMI are independent of the local operator suggests that the scrambling effects washes out the information about the initial state, and the steady state is typical. Additionally, we would like to study how finite system size affects the scrambling ability of the dynamics. ・What we obtained: We found that regardless of the operators and boundary conditions, the late time value of BOMI and TOMI follows Page’s curve. This suggests the scrambling effect washes out the information about the initial state and boundary conditions, so that the late time steady state is typical. We also found that the late time value of TOMI and BOMI can depend on the system size. This suggests that the finiteness of the system prevents the dynamics from breaking nonlocal correlation. ・Motivation for the disordered Heisenberg chain: We would like to study the properties of circuits with weak scrambling effects that preserve the quantum nature of the initial state. As a remarkable example of a circuit that would have such a weak scrambling effect, we considered a spin system with a chaotic and MBL phase. ・What we obtained: In regions of weak disorder strength, regardless of boundary conditions and local operators, we find that the late time value of TOMI and BOMI follows the Page curve, as in Ising model in the chaotic region. In the regions of strong disorder strength, TOMI and BOMI periodically evolve in time. The period is determined by the local interaction. We propose an effective model describing the time evolution of BOMI and TOMI in the Heisenberg model with the weak disorder. In this model, the periodic motion of BOMI and TOMI is caused by the offdiagonal terms when the density matrix is expanded in the zdirectional basis. Thus, this periodic motion may be due to the quantum nature of the state. Another suggestion was that the paper is rather long and should be shortened a bit. We moved many results to the appendices. However, after adding explanations for clarity, the manuscript has become a bit longer than the previous version. If the paper is too lengthy, we will shorten it and would appreciate it if you could point this out to us.
In the following, we respond to your individual questions and suggestions. We hope you will be satisfied with our responses.
 Eq 1  I understand the general idea the authors are trying to convey, but this equation is not correct and neither is the claimed relation between energy and temperature, in general.
Thank you for a useful comment. We have removed the last approximate expression in Eq. (1) as we think it may lead to confusion and was not necessary for the subsequent discussion. To prevent any confusion for the reader, we have also rewritten Eq. (3) to define thermalization of a subsystem.
 Eq 3  missing normalization
Thank you for pointing this out. We have added the normalization constant to Eq. (3).
 Eq 5  can this be reformulated to be clearer?
Thank you for your comment. To make the explanation clearer, we have added the following sentences: “If the late time behavior of observables in V are approximated by their thermal expectation value, then we say that thermalization of V occurs. For example, in addition to the mutual information content studied in this paper, the onepoint function and the distance between states are also used as indicators of thermalization of subsystems. If the sybsystem thermalizes, these observables behave as follows:” We have also changed Eqs. (4,5).
 Eq 9  this quantity is claimed to be “exponentially large”  isn’t it bounded and of order 1 in the models studied here?
Thank you for pointing this out and apologize for the confusion. The part you pointed out was written in such a way that the results from the twodimensional holographic CFT would be confused with the results from the spin system. Therefore, we have rewritten the description to first refer to the behavior of the squared commutation relation in the twodimensional holographic CFT, and then refer to the behavior in the spin system.
 Sec 2.2. “we expect quantum information to be preserved locally”. Generically only classical information is preserved locally in the MBL phase (the zcomponent of the lbit is conserved but not the xcomponent).
We agree with the referee and remove this misleading paragraph.
 p6 “robust against noise” What is meant here? Noise (timedependent fluctuations in the Hamiltonian) will generically destroy the MBL phase.
Thank you for your question. We agree with the referee that timedependent fluctuations in the Hamiltonian will generically destroy the MBL phase. We have changed the sentences to clearly distinguish between spatial and temporal fluctuations: “Also, note that systems with spatially quasiperiodic modulation, whose nonuniform feature is determined by a few parameters as opposed to lacking longrange correlation, can be manybody localized. Even in these cases, we can generally expect the manybody localized phase to be robust against timeindependent weak perturbations.”
 “linear in noninteracting systems or general nonintegrable systems without localization”. This statement assumes translational invariance. Otherwise, sublinear growth is possible, even without localization.
We agree with the referee and add “translationally invariant” between “general” and “nonintegrable”.
 p9 “unless O is the identity operator” This is not correct, there are other O with this property
Thank you for pointing this out. We have rewritten the part as follows: For example, if O(x, t) is the identity operator, then the entanglement structure of the state in (20) is the same as that of maximally entangled state. In the general case, O(x, t) can make the quantum entanglement structure of the state in (20) less entangled than that of the maximally entangled state.
 p10, 11 contain many vague (e.g. “unrelated to each other”, “data regarding the local entanglement structure”) or repetitive statements
Thank you for the comment. We have added Section 2.3.2 and sentences under Eq. 25 on page 11 to clarify the explanation. Also, we have revised the explanation in Page 12 and 13. In Section 2.3.2, we explain that the initial state is given by the direct product state of EPR pairs. Under Eq. 25 on page 11, we explain that BOMI and TOMI are quantities that measure how many of these EPR pairs are lost locally. In Page 12 and 13, we first explain why we study BOMI and TOMI in the case of three configurations: full overlap, partial overlap, and disjoint configuration. The reason is as follows. In 2D CFTs, the effect of scrambling on BOMI and TOMI depended greatly on the way the configuration is set up, so in the spin system we expect that the effect of scrambling on BOMI and TOMI would also depend greatly on the way the configuration is set up. In addition, we have explained how the time evolution of BOMI and TOMI in each configuration can be interpreted in terms of the EPR pairs.
 p19 should S_A,B be I_A,B?
Thank you for the comment. As you said, it should be $I_{A,B}$. Therefore, we have changed to the following: Then, the behavior of $I_{A,B}$ following the plot of Figure 7 is consistent with (30) with $\hat{L} = 2l + s$.
We sincerely hope that these revisions meet your expectations and address the concerns you raised. We are grateful for the opportunity to enhance the clarity and impact of our paper through your valuable input. Once again, we would like to express our appreciation for your time and expertise in reviewing our work.
Author: Eric Mascot on 20230825 [id 3925]
(in reply to Report 2 on 20210512)We would like to thank you for reviewing the paper and providing valuable feedback. We have taken great care to address the “absence of significant theoretical analysis” by thoroughly revising the text to elucidate the significance of our results in a clear and understandable manner. We have made significant changes to the “Summary of results” sections which outlines our findings. Here, we briefly explain the motivation and analysis.
・Motivation for the Ising chain with magnetic field: We would like to study whether the late time evolution of the BOMI and TOMI of local operators depends on the operators in systems with the strong scrambling ability and few degrees of freedom. In twodimensional holographic CFT, systems with the strong scrambling effect and large degrees of freedom have a late time value of BOMI and TOMI that does not depend on the local operator. Since the initial state considered in this paper is characterized by the Pauli operators, the fact that the late time BOMI and TOMI are independent of the local operator suggests that the scrambling effects washes out the information about the initial state, and the steady state is typical. Additionally, we would like to study how finite system size affects the scrambling ability of the dynamics. ・What we obtained: We found that regardless of the operators and boundary conditions, the late time value of BOMI and TOMI follows Page’s curve. This suggests the scrambling effect washes out the information about the initial state and boundary conditions, so that the late time steady state is typical. We also found that the late time value of TOMI and BOMI can depend on the system size. This suggests that the finiteness of the system prevents the dynamics from breaking nonlocal correlation. ・Motivation for the disordered Heisenberg chain: We would like to study the properties of circuits with weak scrambling effects that preserve the quantum nature of the initial state. As a remarkable example of a circuit that would have such a weak scrambling effect, we considered a spin system with a chaotic and MBL phase. ・What we obtained: In regions of weak disorder strength, regardless of boundary conditions and local operators, we find that the late time value of TOMI and BOMI follows the Page curve, as in Ising model in the chaotic region. In the regions of strong disorder strength, TOMI and BOMI periodically evolve in time. The period is determined by the local interaction. We propose an effective model describing the time evolution of BOMI and TOMI in the Heisenberg model with the weak disorder. In this model, the periodic motion of BOMI and TOMI is caused by the offdiagonal terms when the density matrix is expanded in the zdirectional basis. Thus, this periodic motion may be due to the quantum nature of the state.
In the following, we respond to your individual questions and suggestions. We hope you will be satisfied with our responses.
We thank you for your question. Yes, the behavior of TOMI depends on the integrability. More precisely, the behavior depends on the scrambling effects of the dynamics (e.g., arXiv:1511.04021, 1812.00013). The timeevolved states in this paper are initially equivalent to a product of EPR pairs. BOMI is proportional to the number of EPR pairs in the union of A and B. If dynamics do not scramble, this EPR pair picture works well, and the BOMI is zero. Conversely, if the dynamics do have scrambling ability, some of the EPR pairs are locally hidden in the time evolution, so that TOMI becomes negative. A large negative value of the TOMI suggests a strong scrambling ability of the dynamics. We have added this explanation below Eq. 25.
In the chaotic regime, the strong scrambling washes out the information about the local operator or boundary conditions. In this case, we expect the entanglement entropy to follow the Page curve. We have added a paragraph in section 3.2 describing the theoretical understanding.
Operator entanglement entropy does not appear to logarithmically increase in time. Generally, entanglement entropy strongly depends on the initial state, subsystem, and the time evolution operator. When entanglement entropy grows logarithmically in time, the initial state is in an unentangled state, and the subsystem is taken to be half of the total system, as in PRL 109, 017202 and PRB 104, 214202. In this paper, the initial state is given by a product of EPR states, which we have added in section 2.3.2, and the subsystem is not half of the total system. Therefore, the OEE does not grow logarithmically in time. Instead, BOMI and TOMI in MBL is periodic in time (quantum revival). The periodic motion is determined by the strength of the disorder, as described in section 4. This periodicity suggests the state returns to the initial state, and the time evolution operator in MBL may preserve the quantum nature of the state.
We apologize for the confusion caused by the “error” bars. These bars show the minmax range over disorder configurations. We believe the disorder averaged lines show clear trends without the need for additional averaging.
Thank you for the useful feedback. We have moved many of the figures to the appendix. We have added a section describing each configuration shown in Figure 2. In 2D CFT, the effect of scrambling on the time evolution of BOMI and TOMI is highly dependent on the configuration. However, in the spin systems considered in this paper, the full overlap configuration shows a dependence on the scrambling ability. Therefore, we move the partial overlap and disjoint configurations to the appendix.