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Entanglement dynamics with string measurement operators
by Giulia Piccitto, Angelo Russomanno, Davide Rossini
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Submission summary
Authors (as registered SciPost users):  Giulia Piccitto · Davide Rossini 
Submission information  

Preprint Link:  https://arxiv.org/abs/2303.07102v2 (pdf) 
Date submitted:  20230705 15:11 
Submitted by:  Piccitto, Giulia 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approaches:  Theoretical, Computational 
Abstract
We explain how to apply a Gaussianpreserving operator to a fermionic Gaussian state. We use this method to study the evolution of the entanglement entropy of an Ising spin chain following a Lindblad dynamics with string measurement operators, focusing on the quantumjump unraveling of such Lindbladian. We find that the asymptotic entanglement entropy obeys an area law for finiterange string operators and a volume law for ranges of the string which scale with the system size. The same behavior is observed for the measurementonly dynamics, suggesting that measurements can play a leading role in this context.
Author comments upon resubmission
List of changes
1. We updated the figures
2. We added a new section to discuss the role of gamma and r in the entnaglement dynamics
3. We improved the bibliography
4. We corrected the typos
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 4) on 202395 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2303.07102v2, delivered 20230905, doi: 10.21468/SciPost.Report.7768
Strengths
1. The paper investigates the existence of a volume law for finite r in a free fermion system, which is a relevant result.
Weaknesses
1. Lack of a conclusive analysis of the MIPT in terms of gamma and r
Report
In the updated version of the manuscript the authors addressed all the minor changes suggested by the Referees.
In particular, they improved the results presented in the previous manuscript:
i) They added new Fig 1 and Fig 2 with correct numerics
ii) Added Fig 3 and 4
iii) Added a new sec 5.4 regarding the phase transition for r=1
1) For $r=1$, the authors suggest a phase transition between logarithmic and arealaw at $\gamma_c=0.2$. How did they extrapolate the critical point?
2) Is it possible to enlarge the size of the system and perform the FSS?
3) In the log phase it should be possible to observe an asymptotic scaling collapse of the entanglement entropy, $S(l,L)=(c/3) \log((L/\pi) \sin(\pi l/L)) +s_0$. The effective central charge $c$ should be a good indicator for a phase transition between area and log low. Could you extrapolate c from your data?
4) Could you increase the size of the system in Fig 4?
Given the limitation of the numerical analysis presented in the paper and of the new analysis for $r=1$, I would not recommend the manuscript for publication in Scipost Physics in its current form.
Report #1 by Anonymous (Referee 3) on 202377 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2303.07102v2, delivered 20230707, doi: 10.21468/SciPost.Report.7464
Strengths
Same as before
Weaknesses
Same as before
Report
In this revised version, G. Piccitto and collaborators have resolved the minor issues raised by the Referees and added a few comments on the manuscript concerning the major criticisms. They also corrected the numerical calculations as they resolved a bug in their computational implementation.
Nevertheless, the Authors failed to respond exhaustively to the major criticisms of both Referees.
Let me argue this fact below.
1) The only technical advance compared to the previous version of the manuscript in this direction is the discussion of the $r=1$ setup. Let me anticipate that this is the only attempt to solve a major comment consistently.
Here the Authors vary $\gamma$ to suggest a phase transition between logarithmic and arealaw measurementinduced phases.
However, it is clear from both reports (and the seeming scope of the work) that the highlight should be the regime $r>1$.
For instance, they could have discussed the $r=4$ (that already appeared as a value in the text).
How the "critical point" is extracted is also dubious: small system sizes are considered $L\le 48$, and no finite size scaling analysis is presented. Let me remark that, compared to the literature, these sizes largely overestimate the value of the critical point, and the effective $\gamma_c(L\to\infty)$ will be much smaller than $\gamma_c\simeq 0.2$.
Indeed, given the noncommuting nature between various measurements $\{A_j\}$ it is not clear that the phases of the monitored system may be only trivial (area) law, with $\gamma_c\to 0$.
These points are important, especially when the manuscript's new physics is computational with no analytical understanding or explanation is tested against the Authors' findings.
2) Another main comment from both the Referees is the missing analysis varying $r$ and if a critical $r_c$ exists. The Authors argue that this is computationally difficult to access (even for free fermions). Hence they do not present any concrete solution in this direction.
I stress that if any new relevant physics is present, it should be exactly in this direction. But no such analysis is done, and no effective data backs up the suggestive claims like "However, our numerics suggest the existence of a $\textbf{critical value}$ of n = $L/r (n > 1)$, separating a regime in which the asymptotic entanglement entropy does not change with the system size (i.e., $n \gg 1$), to a regime where it exhibits a linear growth with $L $(i.e.,$n \sim 1$)."
For example, no finite size scaling to support these facts.
Lastly, I again remark that a derivation of the effectiveness of the QR decomposition was already present in the literature, cf. the manuscript by Fava et al., Ref. [71] in the revised version.
# Recommendation
Given the above comments and in light of the previous Referees' reports, I cannot recommend the revised manuscript for publication in Scipost Physics.
Still, given the limitation of the computational capability of the Authors and the new analysis for $r=1$, I would recommend the manuscript for Scipost Physics Core if:
1) they soften or remove all suggestive claims regarding the existence of transitions if no further finitesize scaling or supporting data is presented.
2) at least for $r=1$ perform a data collapse/finite size analysis with preferibly some larger system sizes.
# Minor issues:
1) Missing $\beta$ in the line before Eq. 40.
2) Eq. 52, as it stands, is wrong. (You can quickly check using twosite numerics or redoing the analytics). Also, maybe $\sigma^\pm = (\sigma^x\pm i \sigma^y)/2$? The Authors should check all the algebra concerning that part, and spell out the various convention/ choices of JordanWigner and \sigma^\alpha they used.
3) The Labels in the insets of Fig 3 are too small.
Requested changes
See report.
# For publication in Scipost Physics Core
1) they soften or remove all suggestive claims regarding the existence of transitions if no further finitesize scaling or supporting data is presented.
2) at least for $r=1$ perform a data collapse/finite size analysis with preferibly some larger system sizes.
Author: Giulia Piccitto on 20231020 [id 4051]
(in reply to Report 1 on 20230707)
We thank the Referee for their careful reading of our manuscript and for acknowledging changes in the new version.
Providing an exhaustive answer to their major criticisms is a difficult, if not impossible, task to be achieved without further extensive and timeconsuming numerical simulations. For this reason, we eventually followed their recommendation, as explained below.
Recommendation
Given the above comments and in light of the previous Referees' reports, I cannot recommend the revised manuscript for publication in Scipost Physics. Still, given the limitation of the computational capability of the Authors and the new analysis for $r = 1$, I would recommend the manuscript for Scipost Physics Core if:
We have followed the advice of the Referee: we addressed all the issues raised and decided to resubmit the revised manuscript to SciPost Physics Core.
1) they soften or remove all suggestive claims regarding the existence of transitions if no further finitesize scaling or supporting data is presented.
The new version avoids any explicit referencing to the existence of entanglement transitions in our setup. We now refer to qualitative changes in the entanglement behavior, when varying the system parameters.
2) at least for $r = 1$ perform a data collapse/finite size analysis with preferably some larger system sizes.
The analysis for $r = 1$ has been extended by collecting data for larger system sizes (up to $L=80$) and additional values of $\gamma$. In particular, we added Fig. 4(b) showing the asymptotic entanglement entropy, normalized to the logarithm of the system size, vs $\gamma$. This shows that, within error bars, the curves for various sizes are overlapping for $\gamma \lesssim 0.15$, suggesting a logarithmic growth of the entanglement entropy in that region. In contrast, for measurement rates larger than $\gamma \approx 0.15$, the various curves drop, suggesting that the asymptotic entanglement entropy grows slower than $\log(L)$, consistently with the hypothesis of the emergence of an area law.
The above behavior evidences distinct scalings of the entanglement entropy with the system size in the two regions separated by $\gamma \approx 0.15$ (from logarithmic to arealaw). Since we are not able to present a more robust claim of the occurrence of an entanglement transition at $\gamma_c$, through further precise numerics, we removed explicit references to the existence of an entanglement transition.
Minor issues:
1) Missing $\beta$ in the line before Eq. 40.
We have added it.
2) Eq. 52, as it stands, is wrong. (You can quickly check using twosite numerics or redoing the analytics). Also, maybe $\sigma^\pm = (\sigma^x \pm i \sigma^y)/2$? The Authors should check all the algebra concerning that part, and spell out the various convention/ choices of JordanWigner and $\sigma^\alpha$ they used.
The Referee is correct. We have now amended Eq. 52 (now Eq. 53), explicitly leaving the JordanWigner string $K$ in the expression (see also Eq. 49. where $K$ has been defined).
3) The Labels in the insets of Fig 3 are too small.
We enlarged the labels in the insets of Fig.3.
Author: Giulia Piccitto on 20231020 [id 4050]
(in reply to Report 2 on 20230905)We thank the Referee for their careful reading of our manuscript, for recognizing the relevance of our study, and for realizing that the amended version improves the results of the original one.
Still their main concern is the lack of a quantitative analysis for the entanglement transition in terms of \gamma and r. We have tried to fill the gap along this direction, but unfortunately realized that this turned out to be a rather difficult, if not impossible, task to be achieved without further extensive and timeconsuming numerical simulations.
Nonetheless we decided to focus more on the $r = 1$ case and perform additional numerics to refine our analysis there. Namely, we collected data for larger system sizes (up to L=80) and additional values of $\gamma$. We also added Fig. 4(b) showing the asymptotic entanglement entropy, normalized to the logarithm of the system size, vs $\gamma$.
The analysis provided in the novel Fig. 4(b) shows refined results for the crossover point between logarithmic and arealaw behavior (we now prefer to avoid an explicit reference to a phase transition). In fact we observe that, within error bars, the curves for various sizes are overlapping for $\gamma \lesssim 0.15$, suggesting a logarithmic growth of the entanglement entropy in that region. In contrast, for measurement rates larger than $\gamma \approx 0.15$, the various curves drop, suggesting that the asymptotic entanglement entropy grows slower than $\log(L)$, consistently with the hypothesis of the emergence of an area law.
We have now reached L=80, thus enlarging the previous maximum size L=48. A scaling analysis of the curves for various values of L has been performed in the novel Fig. 4
We tried to extrapolate the central charge from our data. However, this analysis did not provide clear a indication of a possible transition (probably because of finite size effects), therefore we preferred to show only the results in Fig. 4.
We now show data for sizes up to L=80 (in the previous version, we limited our analysis in Fig. 4 to L=48).