Spectral anomalies and broken symmetries in maximally chaotic quantum maps

We thank the Referee for their careful reading of our manuscript and detailed feedback, which has significantly helped improve the clarity of our presentation. In particular, we agree that a careful discussion of "macroscopic" symmetries (to the extent that this is possible) is essential to clearly present our results in context. We address this and all other requested changes in our revised manuscript as indicated below.

Here we provide a detailed changelist and response to the Referee's list of requested changes.
1. We thank the Referee for pointing out that the role of macroscopic symmetries was not clearly stated in the previous version. We have now highlighted the nontrivial nature of the correspondence between symmetries and spectral statistics, in particular by explaining why it is essential to consider only macroscopic symmetries, rather than intrinsic quantum symmetries, before Eq. (3). In short, there is no unambiguous way to define quantum symmetries because any quantum map can be expressed in a diagonal structure in the energy eigenbasis with blocks of size $1$. Further, by transforming to a basis where the eigenvectors are real, one can always identify an antiunitary symmetry of any given quantum system. It is only the restriction to macroscopic symmetries (though imprecise) that identifies which blocks are physically relevant. Thus, it does not seem to us that a clear separation between macroscopic symmetries and quantum symmetries is possible, and to our knowledge, has not been made in the literature. Nevertheless, this is not seen as a "practical" obstacle when studying the spectral statistics of individual systems, which is exceedingly common in the literature.

We also explain why these issue may not be transparent in conventional random matrix theory treatments of this problem in a footnote on the same page. This is because in considering random matrix ensembles rather than individual systems, one is able to restrict to quantum symmetries shared by all members of the ensemble rather than the more numerous set of symmetries of an indiviudal system. Studies of individual systems rather than ensembles, such as ours, do not have this luxury, and must face the ambiguity of how to define symmetries.

2. We have now clarified this in our manuscript as indicated above. By the same token as above, Ref. [48] considers quantum symmetries of random matrix ensembles rather than individual systems, and circumvents the issue of separating macroscopic symmetries and quantum symmetries. However, to the extent that one desires to apply the results of Ref. [48] to identify the "quantum symmetries" of an individual system rather than an ensemble, it is faced with the same ambiguity. Indeed, we agree with the Referee that this ambiguity highlights the nontrivial nature of the connections between symmetries and spectral statistics. Specifically, we show that (1) quantum symmetries cannot be unambiguously associated with spectral statistics without reference to some macroscopic behavior (which has been frequently recognized in studies of many-body quantum chaos), and (2) the correspondence between macroscopic symmetries and spectral statistics is not straightforward even in systems with a well-defined classical limit.

3. We thank the Referee for this comment. We have renamed the overall section to "Results" to better reflect its lengthier content, and kept the summary at the beginning of Section 3 as the actual overview of results (now Section 3.1 - Overview of results). The rest of Section 3 contains details of the results, which we have now made self-contained by including definitions where relevant, while the subsequent sections focus on the derivations. Respectfully, we feel that this structure allows a reader interested in our main quantitative results to focus on Sec. 3 (or just 3.1 for a qualitative summary), while a more specialized reader interested in reproducing all the details of our numerics/derivations may consult the later sections for this purpose. To address the Referee's specific concerns, we have added the definition of nearest-neighbor level spacings and mean gap ratio to the beginning of Section 3.2, and rephrased the start of Section 4.2 to reflect the changes.

4. We thank the Referee for noting the lack of clarity in this description. We amended the statement to clarify it was intended only for the non-phase quantizations, and changed it to "many of which relate to powers of the scaling factor $A$ for the non-phase quantizations [Fig. 8(a),(c)]". We note that while some of the large dips may look linear on the plot in e.g. Fig. 8(a), several of these correspond to different values of $A$. In particular, in Fig. 8 (a), it may be easier to see that there are spikes for all colors; however, the ones for $A=2$ are more numerous due to the smaller value of $A$.

5. We had numerically calculated the Frobenius norms for a variety of commutators in Fig. 14 in Appendix A (also in the previous version). To ensure that an interested reader would not miss this figure, we have added a clearer explanation in the main text (new lines 608-609) that one can see and compare this commutator's norm in the bottom left corners of Fig. 14(a-c).

6. We have changed the references to $A$ as the "slope" of the map to calling $A$ the "scaling factor" of the map, and added this description near the beginning of Section 2 (line 120).

7. We thank the Referee for this suggestion. We have added rows to the summary table in Tab. 2 indicating the preserved classical symmetries for the various quantizations, and added an explanation to the caption. We have chosen to add this information in Tab. 2 instead of Tab. 1 due to its close correspondence to spectral statistics. As we write in the new explanation in the caption (see further discussion in Section 4.1), ruling out symmetries is not entirely straightforward, since the commutation relation would need to be ruled out not just for the standard or "obvious" reflection or TR symmetry operators, but also for other operators that can differ by $O(\hbar)$ but still correspond to the classical symmetry in the semiclassical limit. We clarify that the newly added rows in the table fulfill two purposes: identifying the natural or "obvious" symmetry operators, and identifying whether the classical symmetry is reflected in the short range spectral statistics.