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Geometric measure of entanglement from Wehrl Moments using Artificial Neural Networks
by Jérôme Denis, François Damanet, John Martin
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Submission summary
Authors (as registered SciPost users):  John Martin 
Submission information  

Preprint Link:  https://arxiv.org/abs/2205.15095v1 (pdf) 
Date submitted:  20220607 12:36 
Submitted by:  Martin, John 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approaches:  Theoretical, Computational 
Abstract
In recent years, artificial neural networks (ANNs) have become an increasingly popular tool for studying problems in quantum theory, and in particular entanglement theory. In this work, we analyse to what extent ANNs can provide us with an accurate estimate of the geometric measure of entanglement of symmetric multiqubit states on the basis of a few Wehrl moments (moments of the Husimi function of the state). We compare the results we obtain by training ANNs with the informed use of convergence acceleration methods. We find that even some of the most powerful convergence acceleration algorithms do not compete with ANNs when given the same input data, provided that enough data is available to train these ANNs. More generally, this work opens up perspectives for the estimation of entanglement measures and other SU(2) invariant quantities, such as Wehrl entropy, on the basis of a few Wehrl moments that should be more easily accessible in experiments than full state tomography.
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Reports on this Submission
Report #2 by Anonymous (Referee 2) on 20221017 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2205.15095v1, delivered 20221017, doi: 10.21468/SciPost.Report.5912
Report
The authors consider the problem of estimating the entanglement of symmetric multiqubit pure (SMP) states from a given dataset. In particular, they assume to have access to the first few Wehrl moments (WMs) for a number of SMP states (where the latter are randomly sampled from three chosen distributions). Given the WMs up to a given order qmax, and assuming that they correspond to SMP states, the authors provide three methods to estimate their entanglement content as quantified by the geometric measure of entanglement (GME): 1) using the WM ratio of the largest available moments; 2) using an estimation of the limit of the sequence of the WM ratios (which the authors show to converge exactly to the GME); 3) using an artificial neural network (ANN) previously trained on a different dataset (of SMP states randomly sampled from the same distributions as for the test set). By a statistical analysis, the authors show that the latter method is the most powerful, being able to provide accurate estimates (i.e., within 1% of mean relative error) for qmax below 8 and for state composed of up to 10 qubits. From this analysis, the authors conclude that the combination of ANN and the knowledge of WMs is a powerful tool to estimate entanglement measures (and other SU(2) invariant quantities), based on the fact that WMs should be accessible experimentally with relative ease.
The manuscript is well written and organized. In Sec 2 and 3, the authors clearly made an effort to introduce the reader to the theoretical tools used in their analysis, which considerably ease the reading of the result sections. The results are scientifically sound and, in general, sufficient details are given for the numerical results to be reproducible. Also, the results presented are to the best of my knowledge original.
Despite the above, I have major concerns regarding the suitability of the manuscript for the present journal. Specifically, I think that the results presented do not meet the selective criteria for publication in SciPost Physics. In the conclusions the authors claim that their work open "several perspectives" for the experimental evaluation of entanglement measures in SMP states as well as in generic states, beyond SMP and possibly including mixed states. If such a claim were sufficiently supported by evidences I would have no reservation in recommending the manuscript for publication, as at least one of the SciPost criteria would be met ("Open a new pathway in an existing or a new research direction, with clear potential for multipronged followup work"). However, I do not see convincing arguments to support such a claim. In particular:
1) As said, the method is entirely based on WMs. The main motivation that the authors give to focus on a dataset composed of WMs is that they work under the assumption that it is relatively easy to measure them experimentally. In particular the authors claim that it is feasible in experiments to have access to the first few WMs for SMP states composed of several qubits. The evidence that is given to support such a claim is a set of references reported in the introduction: [17, 19, 22, 23]. However, these articles describe only theoretical proposals, and none of them demonstrate experimentally the measurement of WMs. In addition, to the best of my understanding, none of these articles addresses the multiqubit scenario studied in the present manuscript: Refs [19,22,23] concern continuousvariable systems, whereas Ref [17] focuses on a single qudit. Therefore, as said, I do not find in the manuscript sufficient support to the claim that WMs are easy to obtain experimentally, which is the practical motivation that underpins this analysis and, more importantly, the "perspectives" that it might open.
2) In the analysis reported, the WMs are assumed to be known exactly. In fact the analytical exact expression reported in Eq (16) is used in the manuscript. However, in a practical scenario, the dataset of WMs that should be used to feed the ANNs would only be known approximately [assuming that WMs can be somehow obtained, see point 1) above]. The impact of such uncertainty, which will most probably increase for increasing WM order qmax, is not considered in the present study. This should be addressed thoroughly to support the claim of usefulness of the method in a practical setting.
Besides the two major points above, there are some other relevant points that the authors should address to convincingly support their claims:
3) The authors speculate in the conclusions that their method opens new perspectives for generic states, possibly including mixed ones. While I understand that this is a speculation and that the authors do not intend to assert it with certainty, I still think that it would be useful if such a statement were supported by some preliminary evidence. It is in fact unclear to me if or how the ANN would generalize to the mixed case. The only hint at good generalization properties of the SMPtrained ANNs is given in the appendix C, where the ANN performances on a dynamically generated dataset are presented (see fig 10). These are still pure states though.
4) The authors claim that it is easier to measure WMs than perform full tomography (i.e., the approximate reconstruction of the Husimi function or any other representation of the entire state). Notwithstanding the fact that I agree with such a general statement for a generic state, I am not convinced that this is the case for the specific class of states at hand. In fact, SMPs are a very small subset of the set of pure multiqubit states. Therefore, it is reasonable to expect that full tomography will be significantly simpler to perform for SMPs than it is for the general case, via exploiting the symmetry of the states. Certainly, the scaling would not be exponential with the number of qubits. In other words, the approach via WMs might not be necessarily easier than full tomography in this case. The authors should give more arguments to support the contrary.
5) To put this work in context, it would be important to analyze more in details the connection with previous studies, reporting on possible advantages. I refer to the entanglement witness analysis reported previously for Dicke states and SMP states more in general, for example:
 J. Opt. Soc. Am. B 24, 275 (2007)
 New J. Phys. 11, 083002 (2009)
 Phys. Rev. A 83, 040301(R) (2011)
 Phys. Rev. A 88, 012305 (2013)
 Phys. Rev. Applied 12, 044020 (2019)
In conclusion, I think that the present manuscript reports sufficient new results to be published in a specialized journal, however it does not meet the criteria for SciPost Physics. Therefore I cannot recommend it for publication.
Report #1 by Anonymous (Referee 1) on 2022912 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2205.15095v1, delivered 20220912, doi: 10.21468/SciPost.Report.5683
Strengths
1The paper provides a new and innovative approach to study entanglement, i.e., computing the geometric entanglement in symmetric states, which is a highly important set of states, using artificial neural networks.
2It opens many interesting further research questions, some of which are also discussed in the conclusion.
Weaknesses
1Although permutation symmetric states are interesting the authors only consider a very small class of states and pure states only. Experimentally realistic scenarios would however include mixed states as well.
2It is not so clear what the limitations are of this approach, e.g., concerning its application to mixed symmetric states or even to states without any symmetries at all.
Report
In their article the authors study estimation techniques for the geometric measure of entanglement (GME) based on artificial neural networks. They prove strong bounds on the geometric measure in terms of Wehrl moments that in the limit resemble the GME. Wehrl moments have the advantage that they can be estimated in experiments without relying on full tomography. The authors then show how to estimate the GME from the first few moments only using convergence accelerated algorithms and ANNs. They find that the ANNs perform very well in this task even in cases that are very different from the training data (here states generated from spin squeezing).
The manuscript is very well written and concepts are thoroughly explained. I would accept the manuscript after small revisions (see below) based on the fact that it fulfils the acceptance criterion "Open a new pathway in an existing or a new research direction, with clear potential for multipronged followup work".
Requested changes
1 I think a general reader would greatly benefit from a more detailed introduction to ANNs. The idea of convergence acceleration algorithms is nicely explained for instance. I would ask the authors to provide at least a little background information in the general idea of ANNs.
2Directly before section 5 the authors write "This [the saturation of RME with increasing number of qubits N] is probably due to the fact that the entanglement can take more complex forms for a higher number of qubits".
It is definitely true that entanglement can take more complex forms if the particle numbers increases, but I cannot see how this is a probable cause for the saturation. Could the authors explain in more detail why they believe that this causes the RME to stabilize, or is there any, maybe numerical, evidence for this conclusion?
Author: John Martin on 20230112 [id 3232]
(in reply to Report 1 on 20220912)
The referee writes:
"1 I think a general reader would greatly benefit from a more detailed introduction to ANNs. The idea of convergence acceleration algorithms is nicely explained for instance. I would ask the authors to provide at least a little background information in the general idea of ANNs."
Our response
We thank the referee for this suggestion and have now added a description of the basic working principles of ANNs.
The referee writes:
"2Directly before section 5 the authors write ”This [the saturation of RME with increasing number of qubits N] is probably due to the fact that the entanglement can take more complex forms for a higher number of qubits”. It is definitely true that entanglement can take more complex forms if the particle numbers increases, but I cannot see how this is a probable cause for the saturation. Could the authors explain in more detail why they believe that this causes the RME to stabilize, or is there any, maybe numerical, evidence for this conclusion?”
Our response
The referee is right to say that our explanation is not satisfactory as it stands. We believe that for a higher number of qubits, there is a greater spectrum of states with the same first Wehrl moments but different GMEs. This would imply that the input to the ANN is not sufficient to distinguish between these different states and would explain the observed increase in error. We have now modified the discussion in the manuscript accordingly.
The referee writes:
"1Although permutation symmetric states are interesting the authors only consider a very small class of states and pure states only. Experimentally realistic scenarios would however include mixed states as well. 2It is not so clear what the limitations are of this approach, e.g., concerning its application to mixed symmetric states or even to states without any symmetries at all.”
Our response
We appreciate the comments of the referee and have taken our study further by considering some mixed states as well. More specifically, we trained neural networks on Werhl moments of depolarized mixed states for small numbers of qubits (N = 2, 3, 4) in order to predict their GME that we calculated by SemiDefinite Programming. The results are again quite conclusive. They are presented in a new Section 6 and the method of calculating the GME is presented in the new Appendix E, as it may also be of interest to the reader.
Author: John Martin on 20230112 [id 3231]
(in reply to Report 2 on 20221017)Measurement of WM
The referee writes:
"1) As said, the method is entirely based on WMs. The main motivation that the authors give to focus on a dataset composed of WMs is that they work under the assumption that it is relatively easy to measure them experimentally. In particular the authors claim that it is feasible in experiments to have access to the first few WMs for SMP states composed of several qubits. The evidence that is given to support such a claim is a set of references reported in the introduction: [17, 19, 22, 23]. However, these articles describe only theoretical proposals, and none of them demonstrate experimentally the measurement of WMs. In addition, to the best of my understanding, none of these articles addresses the multiqubit scenario studied in the present manuscript: Refs [19,22,23] concern continuousvariable systems, whereas Ref [17] focuses on a single qudit. Therefore, as said, I do not find in the manuscript sufficient support to the claim that WMs are easy to obtain experimentally, which is the practical motivation that underpins this analysis and, more importantly, the ”perspectives” that it might open."
Our response
Firstly, we have not been as firm in our statements as the referee claims when he/she writes that ”The main motivation that the authors give to focus on a dataset composed of WMs is that they work under the assumption that it is relatively easy to measure them experimentally.” In our introduction, we already stressed the purely theoretical relevance of our research question and emphasized the importance of Wehrl’s moments from a purely theoretical point of view. As we mentioned in our Introduction: ”They [Wehrl moments] have been used to define measures of nonclassicality, chaoticity or entropy of quantum states [16, 18, 21], and have some relevance in various contexts, such as for the characterization of quantum phase transitions [20, 21].” The fact that Wehrl moments are experimentally accessible quantities is not strictly essential in our work, although it adds to its motivation. The main objective of our work is to determine theoretically whether the partial information about the state of a quantum system present in its first few Wehrl moments is enough to accurately estimate its degree of entanglement. This question is of relevance even in a theoretical scenario where the calculation of a few Wehrl moments would be more direct than the calculation of the full density matrix. Furthermore, we would like to stress that we have only written that Wehrl moments are experimentally accessible quantities that should be more easily accessible in experiments than the full state tomography, and no more than that. On the other hand, we agree that our claim that it is possible, in experiments, to access the first WMs for SMP states composed of several qubits was a bit hasty in view of the references we cited. We have rewritten part of the introduction to remove this ambiguity and make the purely theoretical interest of our work more obvious.
Advantage over full tomography
The referee writes:
"4) The authors claim that it is easier to measure WMs than perform full tomography (i.e., the approximate reconstruction of the Husimi function or any other representation of the entire state). Notwithstanding the fact that I agree with such a general statement for a generic state, I am not convinced that this is the case for the specific class of states at hand. In fact, SMPs are a very small subset of the set of pure multiqubit states. Therefore, it is reasonable to expect that full tomography will be significantly simpler to perform for SMPs than it is for the general case, via exploiting the symmetry of the states. Certainly, the scaling would not be exponential with the number of qubits. In other words, the approach via WMs might not be necessarily easier than full tomography in this case. The authors should give more arguments to support the contrary."
Our response
Indeed the full tomography of symmetric multiqubit states scales as \(\mathcal{O}[(N+1)^3]\) as was recently shown in M. Perlin et al. Phys. Rev. A 104, 062413 (2021). For a general mixed state, the Wehrl moment of order q = 2 takes the form
where \(\rho_{LM}\) are the state multipoles. Because \(\rho_{00}^2=\frac{1}{N+1}\), in order to determine the Wehrl moment of order 2 of a state, we only need to know the relative importance of consecutive multipoles \(\sum_{M=(L+1)}^{L+1}\rho_{L+1M}^2/\sum_{M=L}^L\rho_{LM}^2\) in numbers scaling as \(\mathcal{O}(N)\), which is of course much less than knowing all the state multipoles in numbers scaling as \(\mathcal{O}(N^2)\). Hence, even for symmetric states, this suggests there is an advantage of using the knowledge of a few WMs over the knowledge of the full state. We have added a paragraph in the main text to emphasize this.
Precision on WM
The referee writes:
"2) In the analysis reported, the WMs are assumed to be known exactly. In fact the analytical exact expression reported in Eq (16) is used in the manuscript. However, in a practical scenario, the dataset of WMs that should be used to feed the ANNs would only be known approximately [assuming that WMs can be somehow obtained, see point 1) above]. The impact of such uncertainty, which will most probably increase for increasing WM order qmax, is not considered in the present study. This should be addressed thoroughly to support the claim of usefulness of the method in a practical setting."
Our response
The uncertainties on the value of the Wehrl moments will depend on the specific approach to evaluate them theoretically or obtain them from experimental results. Here, we have only demonstrated a proof of concept of the use of neural networks for the estimation of entanglement based on Wehrl moments. The wide use of neural networks for nonlinear regression in realworld problems is an indication that they are naturally robust to noise. Nevertheless, at the request of the referee, we have now tested our neural networks on noisy Wehrl moments, and have added a brief discussion in the main text and our conclusive results in an appendix of the revised manuscript.
Class of states
The referee writes:
"3) The authors speculate in the conclusions that their method opens new perspectives for generic states, possibly including mixed ones. While I understand that this is a speculation and that the authors do not intend to assert it with certainty, I still think that it would be useful if such a statement were supported by some preliminary evidence. It is in fact unclear to me if or how the ANN would generalize to the mixed case. The only hint at good generalization properties of the SMPtrained ANNs is given in the appendix C, where the ANN performances on a dynamically generated dataset are presented (see fig 10). These are still pure states though."
Our response
We appreciate the comments of the referees and have taken our study further by considering some mixed states as well. More specifically, we trained neural networks on Werhl moments of depolarized mixed states for small numbers of qubits (N = 2, 3, 4) in order to predict their GME that we calculated by SemiDefinite Programming. The results are again quite conclusive. They are presented in a new Section 6 and the method of calculating the GME is presented in the new Appendix E, as it may also be of interest to the reader.
Other minor points
The referee writes:
"5) To put this work in context, it would be important to analyze more in details the connection with previous studies, reporting on possible advantages. I refer to the entanglement witness analysis reported previously for Dicke states and SMP states more in general, for example: * J. Opt. Soc. Am. B 24, 275 (2007) * New J. Phys. 11, 083002 (2009) * Phys. Rev. A 83, 040301(R) (2011) * Phys. Rev. A 88, 012305 (2013) * Phys. Rev. Applied 12, 044020 (2019)"
Our response
We thank the referee for pointing out previous works on the entanglement of symmetric and permutation invariant states. We have briefly mentioned the links and differences with some of them in the introduction.