Non-abelian symmetry-resolved entanglement entropy

1-Formal and precise analysis of the structure of the entanglement in the presence of non-abelian symmetries.

2-Well-written introduction with a summary of the main notions studied in the manuscript.

Lack of comparison with the existing literature about symmetry resolution and comments on the physical aspects of the problem.

In this manuscript, the authors study in great detail the structure of the Hilbert space, states, subsystems, observables and entanglement in the presence of non-abelian symmetries. They point out an interesting factorization between the algebra of $K$-local observables, that in general are not invariant under transformation of the symmetry group, and $G$-local observables, which are both local and invariant. The $G$-local observables in a subsystem $A$ are associated with a density matrix $\rho_{GA}$ and the authors define the symmetry-resolved entanglement (SRE) as the entanglement associated with this density matrix.

I am slightly confused between the definition of the SRE used in this paper and the one introduced in Ref. [13]. Is it correct to say that, in the framework of [13], the SRE should be associated to $\rho_{GA}^{j_A}$, rather than $\rho_{GA}$? In other words, Eq. 9 shows the decomposition of the entanglement of $\rho_{GA}$ into 2 contributions, known as configurational and fluctuation entanglement, and the SRE is given by $-\mathrm{Tr}(\rho_{GA}^{j_A}\log \rho_{GA}^{j_A})$. The symmetry resolution is usually done with respect to the charge of the subsystem ($j_A$ in this case), not of the total system ($j$).

I find the paper very interesting for people working on entanglement and symmetries, however, in my opinion, even though they authors report a comprehensive set of citations, there are not comparisons of their results with the ones existing in the literature. For instance, how do their results compare with the symmetry-resolution studied in the presence of a $U(1)$ Haar-random ensemble in Ref. [5] (there is only a short sentence in the conclusion)? And with the results for the non-abelian symmetry resolution in critical systems ([13,83,84, JHEP 2023, 216 (2023)])? Another recurrent question in the study of the symmetry resolution is the equiparition of the entanglement (i.e. Ref. [14]). It would be helpful if the authors could comment it, especially in the order of limits they consider here (i.e. with a fixed subsystem fraction). In my opinion, these discussions related more to the physical aspect of the problem could make the results even more interesting for readers who are not familiar with the mathematical language used in this paper.

Therefore, I would recommend the paper for publication only after the following changes.

1-As I have mentioned in the report, I understand that the symmetry-resolution of the entanglement in the presence of a non-abelian symmetry can be defined in terms of the irreducible represesentation of the symmetry group, but the authors should comment the connection between the entanglement of $\rho_{GA}^{j_A}$ and $\rho_{GA}$. Given that the nomenclature 'symmetry-resolution' also enters in the title of the manuscript, it would be appropriate avoiding conflicts with the existing literature.
2-More in general, it would be beneficial commenting more their findings in light of the results about the symmetry resolution of the entanglement.
3- An approach to symmetry resolution in the context of operator algebras has been considered in 2305.02343, where the authors study the resolution of the modular flow of operators belonging to algebras of invariant observables. Would it be correct to claim that the authors of the present manuscript consider an algebraic setup analogous to the one in 2305.02343, focusing on the 'resolution' of a different quantity, namely the entanglement entropy?

Ask for minor revision

1. The paper provides a novel description of entanglement measures in systems possessing an internal non-abelian symmetry.
2. The paper generalises known entanglement measures and the notion of locality, to non-abelian symmetries.
3. The paper provides relevant analytical and numerical examples.
4. The paper introduces a more mathematically rigorous treatment of the problem.
5. The paper is very well written.

1. The applicability of the results to known models could be made clearer.

I read this paper with great interest and I think it is a high quality paper. Clearly a lot of effort has gone into the writing, editing, the quality of figures and presentation overall, which I really appreciate. I think that the paper meets all the requirements for publication in SciPost.

The paper provides a new viewpoint and mathematical approach to defining symmetry resolved entanglement measures in systems with non-abelian symmetries, leaving scope for further generalisations and applications to various models. It proposes a novel treatment and definition of useful entanglement measures in this context, by placing particular attention on the choice of entanglement state and on the definition of locality of the entanglement region. These are all very interesting and deep concepts, in the context of entanglement measures.

The paper is is well written, provides a good literature review and comprehensive set of citations, contains a lot mathematical details as well as explicit examples for specific models. Finally, the Appendix provides further details on the more intricate computations, such those related to the derivation of asymptotic properties.

All in all, the paper could be published as is. I have some questions/suggestions that I present below, but I would consider these minor changes that the authors may want to think about.

As I said in the report, I don't really have a request for any major changes. However, there is something that I noticed which perhaps could be improved and would help make the paper more appealing to people like me, which are less mathematically inclined and more focused on analytical computations.

It strikes me that because most notions in the paper are introduced from the mathematical view point of symmetry groups and symmetry algebras, there is very little discussion of how the physical nature of the system can affect the results. What I mean mainly is, for instance, whether it makes any difference that the system is critical or not. Given that so many studies of this quantity have been done for 1+1D critical and non-critical (mainly integrable) systems I think that it would be useful to highlight how the nature of the correlation length (infinite or finite) affects the results. Where/how does it enter?

For example, the first model that is mentioned is the Ising chain with random couplings. Such a system is not critical in the usual sense but it is known from studies of more standard entanglement measures, that many such measures (on average) scale as in critical systems, with the central charge replaced by an "effective" central charge. See e.g. the review https://arxiv.org/abs/0908.1986. So this chain is a rather special example, even before considering the role of any symmetries.

Publish (surpasses expectations and criteria for this Journal; among top 10%)