Local measures of entanglement structure and dynamics in CFTs and black holes

1-The paper is very clearly written, with a careful and balanced introduction to the concept of entanglement contour and the recently developed technology of entanglement islands.

1-In the particular context used, the entanglement contour reduces to a repackaging of the single interval entanglement entropies that have been derived in previous works, making it somewhat unclear whether genuinely new things are being found by this analysis --an exception to that, however, is Section 5.3, see main Report.

The author explores the so-called entanglement contour, which is intended to be a measure of the ``density'' of entanglement entropy: How much a given subset of the degrees of freedom of a region contribute to its entanglement. They carefully review the ambiguities present in defining this notion and discuss the additional physical conditions that can be introduced to pin it down, in order to motivate a particular definition of the entanglement contour in terms of conditional mutual information (CMI). In the context of single interval entanglement contours in 1+1D QFT that the rest of the work focuses on, this reduces to a derivative of the Von Neumann entropy with respect to the interval's endpoints. In this sense, it is a repackaging of the information contained in the set of interval entropies.

They then explore the behavior of this object, first in quenched CFTs and in local excitations of the CFT vacuum, where they observe an interesting time evolution, and then in the context of 2D AdS black holes in JT gravity coupled to a 1+1D bath CFT in flat space. In the latter case, they see a reflection of the appearance of ``entanglement islands'' in the computation of bath subregion entropy in the corresponding entanglement contour: A jump of its value from O(c) to O(1) at the location where the island transition takes place. This is an interesting observation. It is perhaps somewhat unsurprising, given that we already knew that single interval entropy changes behavior at this point, however the relevant question is whether this behavior of CMI illuminates some new aspect of the island transition.
Section 5.3 attempts to derive such a consequence by pointing out the link between CMI and the fidelity of state reconstruction from its reduced density matrices. In particular, a connection is suggested between the appearance of an island in the entanglement wedge of a radiation interval and the protection of the global state against erasure of the black hole degrees of freedom. This is an intriguing observation, worth further investigation.

Overall, the author is knowledgeable of the subject and careful in their computations and arguments and the material of this work is of general interest for the community so I recommend publication.

1- The paper is well written, well formatted and easy to understand.
2- The paper cointains new results on entanglement entropy and applies them to cases of physical interest. Some interesting insights are uncovered.

1- The generalization to higher dimensions seems not so straightforward.
2- It is still unclear how useful can a local measure of entanglement be, such as the one given by the specific entanglement contour proposal. The conditions are in principle physically motivated, however, entanglement entropy is intrinsically non-local. I wonder if these conditions are restricting the type of microstate in a non-trivial way.

This paper explores in some detail a concrete proposal for partial entanglement entropy and entanglement contours, which are aimed to identify local contributions to entanglement entropy of a subsystem. The paper is well written and contains some novel results, some of which may be of interest to the community and inspire new avenues for future research.

Before recommending this article for publication, I have a couple of questions and possible ideas that could be implemented, which may help improve the overall quality of the manuscript.

1- Section 3: Can the formula (3.9) be applied to higher-dimensions if one consider the case of infinite strips (i.e. when one has translation invariance along the extra dimensions)? There are also known results for CFTs in the vacuum state and it would be interesting to see how the contours depend on the number of dimensions d, at least for cases with the given symmetry.

2- Section 3: I am intrigued about the connection between the entanglement contours based on CMI and bit threads. Section 3.5 deals with the relation of the CMI proposal to kinematic space, i.e., the space of geodesics, in the context of AdS_3/CFT_2. However, it is also well known that there are specific bit thread constructions based on geodesics, e.g., arXiv:1811.08879, which also work in higher dimensions. Can these constructions be used to define analogous "CMI contours" in higher dimensions?

3- Section 4: In 2d CFTs there are other well-known results for the modular Hamiltonian in dynamical cases, including global and local quenches. See, e.g., 1608.01283. I think these results can also be used to construct contours and might yield some interesting physical insights into the quasi-particle picture for entanglement propagation.

4- Section 5: is there any obstruction to consider an eternal two-sided black hole (at finite T) coupled to a bath, as in 1910.11077? I do not understand why one needs to restric to the zero temperature case. Note that from the two-sided case one can also extract a dynamical Page curve, which resembles the evolution of entanglement entropy after a global quench (linear growth and saturation at the Page time).

I would like the author to address points 1-4 in my report (either present explicit calculations or add relevant comments addressing these points). Once this is done I would be happy to recommend the article for publication in SciPost.

1, Contains important and novel results on several aspects of entanglement contour and partial entanglement entropy;

2, Potentially inspires new research directions about entanglement contour and quantum gravity;

3, The paper is easy to understand, very clearly and carefully written;

The discussion on resolving the ordering ambiguities in higher dimensions is not very convincing to me. As was mentioned by the author, this remains to be an open problem.

The entanglement contour (or partial entanglement entropy (PEE)) quantifies how much different degrees of freedom (or sub-regions) within a given region A contribute to the entanglement entropy of A. The PEE is a newly proposed quantity in quantum information, which possesses the special property of additivity. The entanglement entropy is only a number hence hard to capture all the information in the density matrix. The entanglement contour is expected to give a finer description for the entanglement structure of quantum systems and furthermore, to help us better understand holography via quantum entanglement.

Although there have been several important progress in this direction, the study on PEE or entanglement contour is still on a primitive stage. The results in this paper has significantly expanded our understanding of many aspects of PEE. I am very happy to recommend this paper to be published in Scipost Physics in the present form.

I list the main results of this paper and my comments in the following:

1, Previously the PEE in 1+1 dimensions was proposed to be given by a linear combination of certain subset entanglement entropies. Here the author made an important observation that, this linear combination is indeed a conditional mutual information (CMI), hence relate the PEE to a known information theoretical quantity and help us better understand the physical meaning of PEE.

2, It is very interesting that the entanglement contour is used to explore the spatial structure of the Hawking radiation, which gives more information than the Page curve. Using the CMI proposal for PEE, the author showed that large bulk regions become protected from erasure in the island phase. This result is novel and inspires new research direction in quantum gravity.
Also I list the typos I found

3, The paper also studied the entanglement contour for two dynamical system, the splitting quenches and the low energy excited states in 2d CFTs, and gave fine-grained picture of how the entanglement structure evolves in dynamical systems.

There may be a typo in the second line of (4.3), where ``$t \geq x$’’ should be ``$t \leq x$’’.