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Estimation of the geometric measure of entanglement with Wehrl Moments through 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:  (pdf)
Date submitted: 2023-01-12 08:59
Submitted by: Martin, John
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Quantum Physics
Approaches: Theoretical, Computational


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 accurately predict the geometric measure of entanglement of symmetric multiqubit states using only a limited number of Wehrl moments (moments of the Husimi function of the state) as input, which represents partial information about the state. We consider both pure and mixed quantum states. 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 which contain less information than the full quantum state.

Author comments upon resubmission

Dear Editor,

We are very appreciative of the comments and questions we got from the referees, as they helped us improve our manuscript significantly. In this revised version, we have provided substantial addaitional scientific works, not simply rewriting. Noteworthy, we have now included a whole new section (Sec. 6) on how to address the estimation of entanglement in mixed states, a new Appendix E presenting how semidefinite programming can be used for calculating the geometric measurement of entanglement of these states (which is not an obvious task), as well as new Appendix D discussing the requested case of noisy Wehrl moments.

Jérôme Denis, François Damanet and John Martin.

List of changes

• Update of the title
• Update of the abstract, discussion, conclusion, acknowledgements and references to match with the changes in the main text
• Update of the introduction to match with the changes in the main text and to respond to the comment of the second referee about the measurement of Wehrl moments
• Update of Sec. 4.4 to give an introduction to ANNs and a better explanation of the behaviour of ANNs with respect to qmax and N
• Added Sec. 6 on mixed states
• Added Appendix D on noisy Wehrl moments
• Added Appendix E on the computation of the GME for mixed multiqubit symmetric states

Current status:
Has been resubmitted

Reports on this Submission

Anonymous Report 2 on 2023-5-21 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2205.15095v2, delivered 2023-05-21, doi: 10.21468/SciPost.Report.7224


As stated in my previous report, my primary critique concerned the lack of a convincing path towards experimental implementability for the method introduced in this work, as claimed by the authors. I also emphasized that if such a path were provided, I would recommend publication based on the high standard criteria of SciPost Physics ("Open a new pathway in an existing or a new research direction, with clear potential for multipronged follow-up work").

In response to my comments, the authors acknowledged that their previous claims about the experimental accessibility of Wehrl moment (WM) measurements were not supported. They stated in their reply that "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."

Consequently, the authors revised some crucial sentences in their manuscript. For instance, they removed the sentence "Importantly, Wehrl moments are experimentally accessible quantities" from the introduction. Furthermore, in contrast to the initial version, the revised abstract no longer mentions the experimental accessibility of their approach.

Based on this, the authors clarified in their reply that the primary motivation for their work is theoretical in nature: "the main objective of our work is to determine theoretically [...]; "This question is of relevance even in a theoretical scenario where [...]."

I appreciate the authors' effort to clarify that their results are, as they stand, of interest only from a theoretical viewpoint. However, this fact can only reinforce my previous critique. In other words, notwithstanding the originality of their findings, I cannot find convincing evidences that their work can open a new research pathway or meet the high standard of this journal.

To be clear, I do not imply that a theoretical work cannot meet the high standard of SciPost Physics. However, firstly, the original version of this work suggested opening a new research pathway based on the experimental accessibility of the new method -- a claim that has now been withdrawn by the authors themselves, as said. Secondly, as a purely theoretical work, I do not find the method introduced here to be innovative and relevant enough:

- The method has only a limited advantage compared to full tomography for the considered states. The advantage seems to be only quadratic [see for example the reply to point 4) of my previous report].

- There is little to no quantitative comparison with at least some of the vast literature of theoretical methods proposed to estimate (or witness) entanglement for this class of states [see point 5) of my previous report], nor for more general states [see point 3) of my previous report].

In summary, I believe that the present manuscript contains sufficient new results to be published in a specialized journal. In particular, with respect to its previous version, motivations and implications are now more clear. However, it does not meet in my opinion the criteria for SciPost Physics.

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Author:  John Martin  on 2023-07-05  [id 3781]

(in reply to Report 2 on 2023-05-21)

List of changes: - Added Sec. 7 describing a protocol for accessing Wehrl moments experimentally (see - Update of the abstract, introduction, conclusion and references to reflect changes to the main text

Finally, we would like to point out that, compared to the first version of the work we submitted, we have also added discussions on GME prediction (i) based on noisy Wehrl moments and (ii) for mixed states. Each of these discussions indicates that the method of estimating GME with ANNs based on Wehrl moments is a viable method.

Author:  John Martin  on 2023-07-05  [id 3779]

(in reply to Report 2 on 2023-05-21)
answer to question
reply to objection

In response to the referee's main concern, we now propose an experimental protocol for measuring Wehrl moments. It is based on the concept of spherical t-design and works for any state. Therefore, we have now reintroduced our claim that Wehrl moments are experimentally accessible quantities. To our knowledge, we are the first to propose an experimental protocol for measuring Wehrl moments.

The protocol we propose requires a number of Stern-Gerlach experiments that increases as the square of the number of qubits instead of the cube as for full mixed-state tomography. This type of advantage is considered to be limited by the referee. It should be noted, however, that this same type of advantage is obtained by Grover's algorithm (square root advantage), which nevertheless makes it a reference algorithm. We also trained ANNs on the basis of Wehrl moments obtained with spherical t-designs. Our results confirm that ANNs still predict the GME with high accuracy, even when the condition for the exact determination of the Wehrl moments, i.e. t=Nq, is not met. In addition, it appears that some of the information obtained by a Stern-Gerlach experiment is not used in our specific protocol. This leaves room for improvements in the scalability of the measurement of Wehrl moments.

With regard to point 3) of the referee's previous report, we would like to emphasise that we have indeed considered more general states than only pure states as in the first version of this work. More specifically, we considered mixed depolarised states and showed that our approach still works for this type of state. In passing, we have conceived a numerical approach for computing the GME of these mixed states.

Finally, with regard to point 5) of the referee's previous report, we think that the references they mentions are difficult to compare with our approach, which is based on partial state information in the form of moments of functions in phase space. Furthermore, we are already comparing our ANN-based method with convergence acceleration algorithms.

Anonymous Report 1 on 2023-3-13 (Invited Report)


In the revised manuscript the basic concepts behind ANNs are much more clear now. The other points that I raised in my previous report are also addressed satisfactorily. I appreciate that the authors discuss mixed states and the advantage over full tomography in the revised version. Therefore, I recommend the publication of this article.

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Author:  John Martin  on 2023-07-05  [id 3780]

(in reply to Report 1 on 2023-03-13)

We thank again the referee for recommending the publication of the article as it is. We hope that the elements we have just added will make the recommendation even stronger.

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