Local Operator Entanglement in Spin Chains

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 two-dimensional 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 non-local 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 off-diagonal terms when the density matrix is expanded in the z-directional 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.

• Do we expect change in behaviour in TOMI in non-integrable and integrable cases? If yes, why?

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 time-evolved 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.

• 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?

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.

• MBL is associated with logarithmic growth of von Neumann entanglement entropy? How does that effect the results on BOMI and TOMI?

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.

• 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.

We apologize for the confusion caused by the “error” bars. These bars show the min-max range over disorder configurations. We believe the disorder averaged lines show clear trends without the need for additional averaging.

• 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.

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.

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 two-dimensional 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 non-local 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 off-diagonal terms when the density matrix is expanded in the z-directional 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 one-point 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 two-dimensional 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 two-dimensional 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 z-component of the l-bit is conserved but not the x-component).

We agree with the referee and remove this misleading paragraph.

• p6 “robust against noise” What is meant here? Noise (time-dependent fluctuations in the Hamiltonian) will generically destroy the MBL phase.

Thank you for your question. We agree with the referee that time-dependent 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 non-uniform feature is determined by a few parameters as opposed to lacking long-range correlation, can be many-body localized. Even in these cases, we can generally expect the many-body localized phase to be robust against time-independent weak perturbations.”

• “linear in non-interacting systems or general non-integrable 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 “non-integrable”.

• 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.