Crossed product algebras and generalized entropy for subregions

The manuscript contains interesting and important discussions. However, I do not recommend it for publication in its current form.

1- The construction studied by the authors is very natural and general and has potential applications also outside of high-energy physics.

2- The authors consider many physically relevant scenarios.

1- The paper is at time too terse, omitting some details which could be helpful to the non-expert reader.

In a spatial subregion, the algebra of observables does not possess neither a well defined trace nor a density matrix, leading to ultraviolet divergences in the entanglement entropy.
In this work, the authors review and extend a construction that alleviate this problem, the crossed product construction.
Every algebra of observables possesses a group of automorphisms generated by the modular Hamiltonian. By adjoining the action of this group to the algebra, the resulting extended algebra admits a trace and a density matrix. This allows the authors to define a generalised entanglement entropy which is in principle free from UV divergences.

The crossed product construction was already introduced in previous works, in particular [Witten 2021], where it was applied to the thermofield double state. The authors review this construction and extend it to many physically relevant examples. The main novelty with respect to the previous works is that the authors show that the construction is in principle independent of gravity, since it only requires the modular Hamiltonian.

After reviewing the general procedure, the authors apply it to several situations, in particular they consider the vacuum of a relativistic QFT in a Rindler wedge and the one of a CFT in a ball shaped region.
The authors consider also examples in holographic theories. They first study the vacuum state in the AdS/Rindler region, in which the action of the modular Hamiltonian is geometric, and then the case of excited states and of multiple boundary intervals.

The construction studied is very natural and general. While the procedure was already introduced in the literature, the authors demonstrate convincingly that it is independent of gravity. For this reason, this work has potentially vast applicability, even possibly in low-energy physics.

The authors put the paper in the appropriate context, citing the previous works and remarking the main differences between their results and the state of the art.

The paper is well written, however at time the authors omit some details that could be helpful to the non-expert reader. As an example, the modular Hamiltonian used in the construction depends on a specific state, however surprisingly the resulting extended algebra is independent of this state. This was well explained in [Witten 2021], but it is not very clear from this paper.

I recommend the publication of the paper after the authors answer some comments.

1- In Eq. (2.7), the authors show that the states belonging to the extended algebra are given by the tensor product of a state in the original algebra with a function $g$. This function $g$ enters the final result of the generalised entropy in, e.g., Eq. (3.14) for the Rindler wedge and Eq. (3.21) for the ball-shaped region.
This is analogous to the result of [Witten 2021], where the function $g$ was determined to be a Gaussian for the thermofield double state.
In this paper, however, the authors do not discuss how to determine $g$ in the physical situations they consider. In particular, it is unclear if $g$ is fixed or it remains free. The authors could comment on this function.

2- In Eq. (3.22) the authors consider the known result for the entanglement entropy of the vacuum of a 2d-CFT in an interval of length $\ell$ \[S = \frac{c}{3}\log\frac{\ell}{a}.\]The authors claim that the parameter $a$ plays the role of an infrared cut-off. This may simply be a typo, but $a$ is actually an ultraviolet cut-off on small lengths.
A way to see that this is the case is the construction of [Cardy, Tonni 2016]. In that paper, $a \ll \ell$ appears as the diameter of two small circles that have to be removed from the boundary of the interval. This regularisation is needed precisely because the entanglement entropy is UV divergent.
This is a minor point, but it is relevant to the stated purpose of comparing the known Eq. (3.22) with the authors' result in Eq. (3.21), since the generalised entropy (3.21) should be UV finite, differently from (3.22).

3- This last comment is optional, but i hope that the authors may clarify a doubt that arises with respect to excited states in AdS/CFT holography.
At pgs. 24-25 the authors discuss the case of excited states in an holographic large-$N$ CFT. They argue that by considering an excited state (keeping fixed the boundary region), the bulk dual region is modified in such a way as to include the bulk dual relative to the vacuum. As a consequence, the bulk algebra relative to the excited states is larger and contains the one relative to the vacuum.
My doubt regards the boundary algebra. Since the boundary subregion is kept fixed, one could naively expect that the boundary algebra does not change. Therefore it seems that the duality between boundary and bulk algebras does not hold anymore. I hope that the authors could clarify this point.

1- Comment on the role of the function $g$, by either briefly explaing how one could determine it or by making it explicit that it is left free.

2- Under Eq. (3.22), change "$a$ is an infrared cutoff" with "$a$ is an ultraviolet cutoff".