Magic phase transition and non-local complexity in generalized W State

-My main concern is that the authors' arguments do not convincingly justify characterizing the studied transition as a 'magic phase transition'. In fact:

1) As also the authors write "it is conceivable that other combinations of [Pauli] strings, with different physical meanings, could detect this transition. The most natural example that comes to mind for our case is the momentum operator". So wouldn't it be simpler, and wouldn't it have a clearer physical meaning, to use the momentum operator to detect the transition? I understand that momentum is not well-defined in non-translationally invariant systems; however, the only model studied here (Eq. 7) is translationally invariant. In fact, it could be valuable to include results for such a scenario, for instance by considering a disordered version of Eq. 7.

In general, is the use of SRE to detect what essentially amounts to the emergence of excitations with finite and opposite (intensive) momenta somewhat overstated? Can simpler observables capture the same phenomenon?

2) Since the SREs are a sort of 'participation entropy' (in the Pauli basis), I would actually be interested in knowing whether other participation entropies can also detect the studied transition. I am referring, for example, to participation entropies in the computational basis, i.e., the inverse participation ratios. Are you aware if that is the case?

3) In general, it seems to me that the authors do not provide sufficient arguments to support the claim that SREs are in any way special compared to a generic combination of Pauli strings. Moreover, if one intends to refer to the transition as a magic transition, it might be useful to investigate the behavior of other magic quantifiers, beyond the SREs. I am referring, for example, to the robustness of magic, or similar monotones (even though these are limited to small system sizes).

-The text is at times difficult to follow, occasionally resembling a stream of consciousness rather than a structured argument. I recommend that the authors reconsider the overall structure of the manuscript and, at the very least, introduce a clear division into sections. This currently missing element is essential to improve the readability. So for instance: Introduction, Model, Results, Conclusions, etc.

-I am not sure I understood Eq.8 and the next sentences. The symbol \mathcal{T} is undefined, so it is very difficult to understand why Eq.8 represents kink states.

1) Introduce Sections and improve readability 2) Provide more compelling evidence to support the claim of a "magic phase transition".

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1. This work discovers the first instance where nonstabilizerness can detect a phase transition, when entanglement fails.

1. Analysis is too simplistic
2. The general utility of the finding is not clear

This work studies the nonstabilizerness, quantified by the stabilizer Renyi entropy (SRE), in topologically frustrated systems, which can be mapped to the generalized W state in some parameter regime. Through this mapping, the authors show that the SRE displays a jump at a quantum phase transition associated to mirror symmetry breaking. This behaviour is in contrast to the entanglement which remains continuous. As such, this is the first instance of a transition which can only be detected by the SRE (however, see below). These findings were numerically illustrated in the 1D XYZ model with frustrated boundary condition.

While the finding itself is interesting, I am not convinced with its general utility. The authors have mentioned the momentum operator as another possible probe of the transition, which is however only well-defined in the presence of translational invariance (TI). The authors thus argued that the SRE is potentially more useful for general systems. However, from the analysis of the generalized W state, it appears that the jump in the SRE is inherently connected to the finite momenta, where the momenta is only well-defined with TI. It is thus uncler if such a transition in SRE also appears in systems without TI. The present analytical and numerical analysis concern a system with TI, where the momentum operator should also work, and easier to compute than the SRE. Therefore, the authors still cannot claim that the transition can only be detected by the SRE. In order to corroborate the generality of the SRE, the authors should provide some examples of systems without TI, and potentially provide numerical analysis on those systems. For example, can the TI in the present model be broken somehow in a way that preserves the phases and their transitions?

I do not think that this work provides a novel link between different research areas, since the connection between nonstabilizerness and frustrated systems as well as the W state has been discussed previously by some of the authors in their previous Scipost publication (Ref. [24]). It may still open a new pathway if the concern above has been appropriately addressed, as it is currently unclear if the SRE really has potential for general use beyond other quantities as the authors claim.

Other comments: - In Fig. (3), the authors show the SRE difference just before and after a point, which is somewhat unphysical. This difference is connected to the derivative of the SRE, which should display discontinuity. I suggest the authors to also show the derivative, which has previously been showed to be an indicator of a transition (e.g. Ref. [21]) - The discontinuity in SRE appears in the subleading term, which is typically nontrivial to extract. Can the authors comment on whether this term can be extracted through specific linear combinations? - Furthermore, can the authors comment on the universality of this term? - Minor: the abbreviation "AFM" is not defined

1. Clarify the generality of the SRE in systems without translational invariance
2. Clarify the nature of the subleading term and its universality

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