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Magic phase transition and non-local complexity in generalized W State
by A. G. Catalano, J. Odavić, G. Torre, A. Hamma, F. Franchini, S. M. Giampaolo
This is not the latest submitted version.
Submission summary
| Authors (as registered SciPost users): | Alberto Giuseppe Catalano · Salvatore Marco Giampaolo |
| Submission information | |
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| Preprint Link: | scipost_202509_00040v1 (pdf) |
| Date submitted: | Sept. 23, 2025, 2:49 p.m. |
| Submitted by: | Salvatore Marco Giampaolo |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We employ the Stabilizer Renyi Entropy (SRE) to characterize a quantum phase transition that has so far eluded any standard description and can thus now be explained in terms of the interplay between its non-stabilizer properties and entanglement. The transition under consideration separates a region with a unique ground state from one with a degenerate ground state manifold spanned by states with finite and opposite (intensive) momenta. We show that SRE has a jump at the crossing points, while the entanglement entropy remains continuous. Moreover, by leveraging on a Clifford circuit mapping, we connect the observed jump in SRE to that occurring between standard and generalized $W$-states with finite momenta. This mapping allows us to quantify the SRE discontinuity analytically.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Author comments upon resubmission
We thank both the two referees referee for their careful reading of our manuscript and for the beneficial criticisms. We appreciate the opportunity to clarify our perspective regarding the universality of the stabilizer Rényi entropy (SRE) as a diagnostic tool.
Hereafter, we present a point-by-point response to each of the issues mentioned below.
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Referee 1
1) The referee is correct to note that, for translationally invariant systems, the momentum operator naturally probes the transition. As we discuss in the rewritten manuscript, however, the SRE remains when translation invariance is explicitly broken. To demonstrate that, we have extended our analysis to a locally defective analog of the XY model with broken translation symmetry. The computations, presented in Appendix C, show that while the flaw qualitatively alters the nature of the transition—increasing the ground-state degeneracy below the critical point and replacing it with a dense succession of crossovers—the SRE still has a discontinuity at the critical point. This establishes that the SRE is not just associated with the presence of a well-defined momentum quantum number, but instead establishes more significant alterations to the ground-state structure. We respectfully disagree, therefore, with concern that the transition seen by the SRE is exclusive to translationally invariant systems.
Although the momentum operator is undoubtedly simpler to compute under the models of the main text, its ubiquity does not transfer outside this setup, as it is no longer a well-defined observable when translational symmetry is lost. The SRE, by contrast, is always well-defined and continues to signal the phase transition even when translational symmetry is lost.
We thank the referee for this comment and the opportunity to clarify the statement regarding the originality of our work. It is indeed true that in our previous paper (Ref.[24]) we have already addressed the role of SRE in topological frustration systems, where it provided valuable insights into the connection between nonstabilizerness and frustration and its correspondence with generalized W states.
However, the present manuscript does make a significant improvement. Here we extend the analysis to topologically frustrated systems in which there is a transition involving a change from a non-degenerate to a two-fold degenerate ground state. We show that the SRE has a sharp discontinuity through the transition, even when more traditional diagnostics such as the entanglement entropy are smooth. This new situation highlights the power of the SRE to locate singular quantum phase transitions that cannot be accessed by usual tools.
We would like to thank the referee for her/his useful remark. In our example, however, the derivative of the SRE is not a good detector of the transition. This is because the phenomenon has nothing to do with a classical continuous evolution of the ground state but with a crossover from one qualitatively different regime to another: a single ground state and a two-fold manifold formed by states with opposite, non-zero momenta.
Thus, as a consequence, the derivative of the SRE is smooth at the transition, while the discontinuity appears explicitly in the SRE itself. Thus, we particularly focused on highlighting the SRE difference before and after the transition point, which perfectly captures the abrupt change in ground-state structure. See also ref. [29]
Thanks to the referee for calling our attention to this fundamental observation. As we detail in the updated manuscript, the discontinuity of the SRE is in fact due to the contribution at subleading order, but not the leading term that is simply extensive and linear in L. More generally, and at least in the short-range one-dimensional case, we have that the SRE and other extensive quantum resources can be separated in the thermodynamic limit into two terms: a leading extensive term and a subleading one connected with the topological structure of the ground-state manifold.
Therefore, an investigation of quantities such as that in Eq.(13) can allow us to distinguish between the subleading term and the leading term, which we also demonstrate in the amended text. For what the referee asks in the second half of his question, while our present results indicate that this subleading term contains universal information regarding the ground-state topology, a full characterization of its universality class is not within the scope of this paper. We consider this an interesting direction for further research and plan to follow it up in detail in the future.
Referee 2
We thank the referee for her/his valuable comment. In fact, in translationally invariant systems, the momentum operator is naturally a marker of the transition. However, as discussed in the new paper, its definition breaks down at once as soon as we give up translation invariance. Thus, we focused on the SRE, which is still well-defined even in non-translationally invariant situations. To satisfy the referee's demand, we explicitly tested this robustness by examining a model with a local defect that breaks translation symmetry. The results, presented in Appendix C, are that while the defect qualitatively alters the nature of the transition, removing the ground-state degeneracy below the critical point, the SRE still shows a sharp discontinuity, therefore invariably detecting the phase transition. This serves to highlight the usefulness of using the SRE rather than the momentum operator as a more general diagnostic.
Also, the results in Appendix A illustrate that, in the thermodynamic limit, any observable expressible as a finite sum of Pauli strings necessarily fails to detect the transition. Thus, even if one were to posit different observables might in practice be able to in fact detect it, their calculation will likely be of comparable computational cost as for the SRE.
We thank the referee for bringing this fascinating and nontrivial point to our attention. It is easy to observe from Eq.(2) that the generalized W states differ from the standard version by relative phases in the coefficients. Since participation entropies depend solely on the moduli of these coefficients, we would naively expect them to be insensitive to the phase transition considered here. This image may be changed, however, if one looks at the system in Eq.(7), or rather its translation-invariant counterpart. There, then, the calculation of the participation entropies exactly is only feasible with information on all the diagonal elements of the reduced density matrix, which restricts calculable thinking to relatively modest system sizes. This limitation can be addressed by the employment of new methods like those newly presented in Kožić & Torre (2025) (https://arxiv.org/abs/2502.06956), and is thus an interesting direction for future work.
We thank the referee for her/his remark. In this matter, however, we should like to clarify our position respectfully. In the revised manuscript, we report evidence that the SRE may detect the transition not only when there is spatial translation invariance but even when it is explicitly broken. Since the SRE is an actual observer of quantum magic, we don't think that investigating other quantities which are extremely closely related would contribute very much to the physical picture already in the paper. All the same, we fully agree with the referee that exploring complementarities in diagnostics is of potential interest. Here, a line of research that is extremely promising is the exploration of quantities different from, yet still conceptually related to magic, such as the nonlocal magic. We have begun a first-principles study along these lines already; technically the research on nonlocal magic is extremely challenging, though, and our current results, for extremely small system sizes, are not robust enough yet to make decisive predictions about its behavior across the entire transition. For that reason, we believe it is better to leave this vital question to future research, where it can be treated in the extent it merits.
List of changes
All the text that was changed in the paper is in magenta.
The main changes are
1) add a new appendix (Appendix C) about our results on systems in which the invariance under spatial translation is violated 2) add sections 3) add several paragraph in the main text to follow the suggestion rised by the two referees (Mainly section 3 and 4)
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 2) on 2025-10-8 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202509_00040v1, delivered 2025-10-08, doi: 10.21468/SciPost.Report.12091
Report
I believe that, thanks to these improvements, the paper is now significantly stronger. While I remain somewhat unconvinced about the broad generality of the conclusions (since I suppose they depend on the specific model considered), I think the work is suitable for publication in SciPost.
Minor remarks:
-I think naming Section III "MAGIC PHASE TRANSITION IN A ONE-DIMENSIONAL FRUSTRATED ANTIFERROMAGNET" would be more honest and accurate, as it is the only model considered there;
-I think readability could be further improved by removing some unnecessary acronyms (now there are a lot: TF, SRE, gWs, AFM). By the way, AFM is defined but then never used;
-I am not sure I understood to which range of Hamiltonian parameters does the "frustrated phase" the authors speak about corresponds. I think it is never mentioned explicitly;
-I am not sure I understood Author's reply to my previous comment 2), in particular from the sentence: "This image may be changed, however, if one looks at the system in Eq.(7), or rather its translation-invariant counterpart." Eq.(7) looks already explicitly translation-invariant. Then the Author mention that "diagonal elements of the reduced density matrix" are required, but when I was speaking about inverse participation ratios (IPRs) I meant IPRs computed from the full, global wave function, and quantifying its spreading / delocalization over a fixed basis (like the z-basis).
Recommendation
Publish (meets expectations and criteria for this Journal)
Report #1 by Anonymous (Referee 1) on 2025-10-2 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202509_00040v1, delivered 2025-10-02, doi: 10.21468/SciPost.Report.12051
Report
First of all, I commented in my previous report that this work does not provide a novel link between different research areas, since it has already been established in Ref. [24]. The authors replied that this work does make an improvement over Ref. [24], and explained the reason. I think this reply misses the point since making an extension/improvement over a previous work, although could be valuable, is not the same as making novel connections between different areas. Therefore, I stand by my assessment that the first Scipost expectation is not met.
Next, the authors provided new results on the XYZ system with defect, which breaks the translational invariance. It turns out that the nature of the transition is changed with the introduction of the defect. In fact, the system displays an unconventional phenomena where there are many crossing points below the transition point. I am not sure if this can still be called a "transition". The original phase transition is associated to mirror symmetry breaking, and since the defect also breaks mirror symmetry, I am not sure that a proper phase transition remains.
In any case, the authors show that the SRE still exhibits discontinuity in the system with defect. However, they did not make any comparison with other quantities that have been known to probe the transition (in the original model). Based on the known properties of the XYZ model (which was studied in detail in Ref. [29]), I suspect that these crossing points occur between states with different momenta (while the momenta is not well-defined anymore, it may still occur approximately). Therefore, I expect that the expectation value of the translation operator, which can still be computed also without translational invariance, would also show discontinuities. Furthermore, it is known that the chirality operator also probes the transition, and one can argue that it is an even more natural quantity since it acts as an order parameter for the mirror symmetry breaking . I would similarly expect that the chirality operator would still probe the "transition" in the system with defect.
I think that the authors overstate the comparison with entanglement, while neglecting a direct comparison with more conventional phase transition tools, such as the order parameter. As such, the authors fail to convincingly demonstrate that the SRE offers an advantage over these standard tools.
This work certainly has some interesting results which warrant publication, but I do not think that it meets the Scipost expectations. I would thus recommend its publication in Scipost Physics Core.
Recommendation
Accept in alternative Journal (see Report)