SciPost logo

SciPost Submission Page

Estimation of the geometric measure of entanglement with Wehrl Moments through Artificial Neural Networks

by Jérôme Denis, François Damanet, John Martin

This Submission thread is now published as

Submission summary

Authors (as registered SciPost users): John Martin
Submission information
Preprint Link:  (pdf)
Date accepted: 2023-10-18
Date submitted: 2023-07-08 19:57
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. We also provide an experimental protocol for measuring Wehrl moments, which is state-independent. More generally, this work opens up perspectives for the estimation of entanglement measures and other SU(2)-invariant quantities, such as the Wehrl entropy, in a way that is more accessible in experiments than by means of full state tomography.

Author comments upon resubmission

Dear Editor,

We thank the referees for their reports and comments. In this second revision of our work, we now address the first referee's main concern regarding the experimental implementability of the method and the experimental determination of Wehrl moments. We also respond to all other more minor points.

All our additional work compared with the first version shows that the method of estimating GME using ANNs based on Wehrl moments is a viable method.

Sincerely yours,
J. Denis, F. Damanet, and J. Martin.

List of changes

- Added Sec. 7 describing a protocol for accessing Wehrl moments experimentally
- 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.

Published as SciPost Phys. 15, 208 (2023)

Reports on this Submission

Anonymous Report 3 on 2023-10-9 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2205.15095v3, delivered 2023-10-09, doi: 10.21468/SciPost.Report.7920


I am now satisfied with the response provided by the authors. It appears that the misunderstanding in the last round of the refereeing process was due to my misinterpretation of the manuscript notation, and for that, I apologize. To enhance clarity, I would suggest that the authors explicitly state in words that the operation involved is non-entangling.

I am now convinced that the protocol introduced in the latest version of the manuscript involves measurements that are not only theoretically feasible but, more importantly, of practical relevance. As a point of fact, this effectively addresses the primary concern I raised in my very first report, particularly regarding the claimed experimental ease of measuring Wehrl Moments (WMs). Specifically, in my first report, I pointed out that I did "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 the "perspectives" that it might open". These concerns are now dissipated.

Therefore, given the latest clarification, the significant changes in the manuscript during the refereeing process, and the substantial progress made in addressing the concerns, I can now recommend the publication of the present manuscript.

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
  • formatting: -
  • grammar: -

Anonymous Report 2 on 2023-8-10 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2205.15095v3, delivered 2023-08-10, doi: 10.21468/SciPost.Report.7636


I would like to provide a brief summary of the evolution of this manuscript up to its current third version. In the initial submission, the authors asserted the experimental accessibility of Wehrl moments ("Wehrl moments are experimentally accessible quantities", refer to Section 1 of the first version). After my substantial critique of this assertion, the second version of the manuscript saw the authors retracting this claim, acknowledging that it "was a bit hasty" (see authors' response to my first report). However, in this latest iteration, they have reintroduced this claim based on their assertion of introducing "a protocol for accessing Wehrl moments experimentally" (see "List of changes" in response to my second report). My present report will focus only on this last point (see my previous reports for further comments).

According to the authors, this protocol is briefly elucidated in Section 7.1 of the revised manuscript. It is based on the evaluation of the Husimi function at a specified number of phase-space points (directions), depending on the degree of the moment to be determined and the system's size.

Unfortunately, the details of this protocol are essentially contained in the very brief passage -- six lines of text -- following the unnumbered equation subsequent to equation (41). The explanation boils down to the following points: (1) In principle, the Husimi function at a particular point can be evaluated by performing a Stern-Gerlach experiment on the multiqubit state after an appropriate rotation; (2) Referring to atomic systems, this task can be achieved following the approach detailed in Ref [42].

Regarding the general case (1), I find some doubts about the effective viability of the protocol. Firstly, the necessary rotation is a highly entangling unitary transformation that transforms a separable state (denoted as |\Omega> in the manuscript) into a multiqubit Dicke state. This process is not straightforward experimentally; for example, in terms of quantum circuits, it will require a relatively long sequence of two-qubit entangling gates. Secondly, the Stern-Gerlach measurement on a multi-qubit state necessitate a setup akin to that employed for full tomography.

As for the reference to atomic systems (2), I note that Ref [42] is a publication from 2000, which has garnered minimal attention within the experimental community to the best of my knowledge.

In essence, the concise portrayal provided by the authors on this critical point fails to persuade me regarding the practicality of their introduced experimental protocol for Husimi function measurements, let alone subsequent Wehrl moment evaluation --- which requires the measurement of the Husimi function at a number of points that increases quadratically with the size of the system and the degree of the moments.

To be clear, let me explicitly mention: I do not have, nor have I ever had, reservations regarding the potential experimental estimation of Wehrl moments "as a matter of principle" (such as through full tomography). Naturally, my apprehensions pertain to their "practical" attainability.

Additionally, I acknowledge the perspective of Referee 1, which highlights the general merit of the manuscript's motivation. I concur with the notion that exploring novel methods for extracting valuable information from high-dimensional systems while bypassing the extensive measurements required for full tomography is worthwhile. The motivations underpinning the manuscript are not areas of contention for me, and have never been.

In conclusion, I am unable to recommend the publication of this manuscript within this journal. The full motivations of my stance are expressed in my previous reports. Regarding this last iteration of the review process in particular, my position rests on the authors' inability to convince me of the experimental viability of their proposed approach.

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
  • formatting: -
  • grammar: -

Author:  John Martin  on 2023-08-24  [id 3924]

(in reply to Report 2 on 2023-08-10)
reply to objection

Firstly, we would like to thank the referee for his/her time and assessment, which undoubtedly helped us to improve our work. We would like to present a brief summary of how we have responded to all the criticisms made by the referee in his/her previous reports, which ultimately led to the third version of our manuscript. We also wish to express our strong disapproval of the argumentation given by the referee in his/her third report regarding the experimental viability of our approach, which, as we explain below, is clearly based on an erroneous statement.

In his/her first report, the referee raised two major concerns : (1) the lack of evidence that Wehrl moments (WMs) are relatively easy to measure experimentally and (2) the fact that WMs would only be known approximately. He/she also noted ”some other relevant points that the authors should address” : (3) how the ANN would generalize to the mixed states case, (4) the fact that the approach via WMs might not be necessarily easier than full tomography for symmetric multiqubit pure (SMP) states and (5) that we should analyze more in details the connection with previous studies.

In our first revised manuscript, we dealt in depth with points (2), (3) and (5). By introducing Gaussian noise, we showed that training an artificial neural network on noisy Wehrl moments can still provide a good estimate of the geometric measure of entanglement. We also generalized our method for mixed states, showing its effectiveness up to 5% loss of purity. In addition, for certain decoherence channels such as the depolarisation channel, we showed that our method still works even with a high loss of purity. Finally, we have incorporated the references recommended by the referee which we considered to be relevant, and pointed out that comparisons with certain previous works were sometimes difficult due to differences in the physical quantities studied. With regard to points (1) and (4), as we did not have a clear protocol to measure the first Wehrl moments at that stage, we acknowledged 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”. As a result, in the second version of the manuscript, we have removed certain sentences from the introduction to eliminate any ambiguity. In his/her second report, the referee considered that ”the manuscript still did not meet the SciPost Physics criteria” because the question of the experimental accessibility of the Werhl moments was still open.

We then continued our efforts and finally found a protocol that addresses point (4) raised by the referee, which we present in the 3rd version of the manuscript. However, it seems that the referee is not convinced by our protocol, his/her main criticism being as follows: ”I have some doubts about the actual viability of the protocol. Firstly, the necessary rotation is a highly entangling unitary transformation that transforms a separable state (denoted by |Ω⟩ in the manuscript) into a multiqubit Dicke state. This process is not straightforward experimentally; for example, in terms of quantum circuits, it will require a relatively long sequence of two-qubit entangling gates. Secondly, the Stern- Gerlach measurement on a multi-qubit state necessitate a setup akin to that employed for full tomography”.

Let us now explain that his/her doubts about the viability of our protocol are totally unfounded, because they stem from a fundamental misunderstanding: the unitary transformation that must be applied on a state ρ in our protocol is in fact non-entangling, being only a spin rotation, equivalent to a symmetric local unitary transformation. The reason is that the multiqubit Dicke state |D_N^(0)⟩ is itself separable, unlike the Dicke states |D_N^(k)⟩ with k different from 0 and N. Indeed, we have that |D_N^(0)⟩ = |Ω0⟩ with Ω0 = (0, 0). We recognise that this is not explicitly mentioned in the manuscript and are willing to add it, i.e. make it clear that |D_N^(0)⟩ = |Ω0⟩ with Ω0 = (0, 0) and |Ω⟩ = R(Ω)|Ω0⟩ = R(Ω)|D_N^(0)⟩. In any case, this observation completely invalidates the first part of the referee’s argument. In addition, measuring the Husimi function by determining the probability that a multiqubit state is in different pure separable states using local rotations is a fairly common technique, see e.g. Refs. [1-3]. As stated by G. S. Agarwal in Ref. [1], this is a protocol which is feasible for single spin systems, collections of two-level systems and even for light polarization. In fact, it has already been routinely implemented in several experiments [4-6], e.g. using half-wave plates and polarising beam splitters in the case of multiphoton polarization states [6]. Regarding the second part of the argument, let us stress that the only way to access information about a (multi-qubit) quantum state is to carry out a measurement, and that’s precisely what a Stern-Gerlach experiment can do! This type of measurement is ubiquitous, so it’s not surprising that it also appears in full tomography protocols, which have already been implemented experimentally. The two doubts raised by the referee therefore in no way call into question the possibility of implementing the protocol we are proposing, using the same techniques as in the above-mentioned experiments.

The main advantage of our protocol lies in its N^2 scaling, as opposed to the N^3 scaling associated with full tomography. This point is clearly acknowledged by another referee who writes ”Concerning the improvement in scaling with N^2 over N^3 for full tomography I want to add here that for many classical computational problems even the slightest improvement in the exponent makes an arguably huge difference.” and this in our view convincingly addresses point (4) of the referee. Therefore, we maintain that our Wehrl moment measurement protocol is experimentally viable and offers an advantage over full state tomography, thus addressing both points (1) and (4) of the referee. In addition, we emphasise that our protocol uses only a partial subset of the information that can be obtained with a Stern-Gerlach device. Although our approach may have limitations, it provides a clear pathway for future improvements and prospects for further studies.

The referee also repeatedly criticised the brevity of the presentation of our protocol, describing it as ”very brief”. We are willing to expand this section further and add additional references if deemed necessary. However, we do not consider that an argument of brevity is such as to call into question the scientific validity and relevance of our proposal. In addition, we think it is important to note that our new section on measuring Wehrl moments spans almost two pages and that the central idea is not limited to ”six lines of text following the unnumbered equation subsequent to equation (41)” as stated by the referee, but is based on the key concept of spherical t-design introduced even before Eq. (41).

In conclusion, we are convinced that we have responded to all the relevant and scientifically well-founded criticisms raised by the referee during the review process. This has improved our manuscript considerably and we are grateful to the referee for this. We are ready to add further details to the description of the protocol if this is deemed necessary, and with this we hope that our work can be published in SciPost Physics based on the criteria ”Open a new pathway in an existing or a new research direction, with clear potential for multipronged follow-up work”.

Sincerely yours,
J. Denis, F. Damanet & J. Martin

[1] G. S. Agarwal, Phys. Rev. A 57, 671 (1998).
[2] B. Koczor, R. Zeier and S. J. Glaser, Phys. Rev. A 101, 022318 (2020).
[3] M. A. Perlin, D. Barberena, and A. M. Rey, Phys. Rev. A 104, 062413 (2021).
[4] R. Schmied and P. Treutlein, New J. Phys. 13, 065019 (2011).
[5] T. Chalopin et al., Nat. Comm. 9 4955 (2018).
[6] C. R. Müller et al., New J. Phys. 14, 085002 (2012).

Anonymous Report 1 on 2023-7-20 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2205.15095v3, delivered 2023-07-20, doi: 10.21468/SciPost.Report.7543


Recently there has been a huge effort to find new ways to learn properties of quantum states without performing full tomography. Moreover, advanced machine learning techniques have been used to handle the large amounts of data generated by experiments with many qubits. Along this direction the authors clearly demonstrate that already a few Wehrl moments can be used to extract relevant information, in this case bounds on the geometric measure, of a state using ANN. They demonstrate that their technique is applicable to mixed states, and they extended it to noisy Wehrl moments, which is of practical relevance. Moreover, in the revised version they devise a method to measure Wehrl moments in experiments based on Stern-Gerlach experiments, which I would assume to be experimentally feasible. Concerning the improvement in scaling with N^2 over N^3 for full tomography I want to add here that for many classical computational problems even the slightest improvement in the exponent makes an arguably huge difference. Altogether, the results are definitely novel, and leave enough room for further investigations. For instance, I believe that the procedure to measure Wehrl moments is not optimal and can still be improved, as the authors also state themselves. Moreover, I wonder if it can be extended to mixed permutation invariant states.

Based on that I still recommend to accept the paper based on the fact that the authors “Open a new pathway in an existing or a new research direction, with clear potential for multipronged follow-up work”.

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
  • formatting: -
  • grammar: -

Login to report or comment