Local measures of entanglement in black holes and CFTs

Report 2 Comment 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.

Clarification added to introduction: "In higher dimensions there are a few finely-tuned examples with sufficient symmetry, such as when A is a ball or infinite strip in the Minkowski plane, that the entanglement contour is effectively one-dimensional which makes the contour well-defined and calculable [5]. In the general non-symmetric higher dimensional case the formula (1.2) does not straightforwardly generalise."

Report 2 Comment 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?

Clarification added to introduction: " In examples where there exists a special set of bit threads based on geodesics [10] the boundary bit thread flux density has been calculated and shown to equal the entanglement contour calculated with the formula used in this paper [7, 11]."

Report 2 Comment 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.

Response: I agree that a natural and interesting direction for future work would be to consider the entanglement contours of perturbations about other states whose modular Hamltonian are known, which include quenched states with time-dependent modular Hamiltonians as the referee mentions.

Report 2 Comment 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).

Clarification added to introduction: "We can calculate the entanglement contour of radiation in any 2d model of black hole evaporation, if the entanglement entropies of the intervals used in the contour formula are known. We explicitly calculate entanglement contour of a non-gravitational bath coupled to a zero temperature AdS2 black hole, which is one of the simplest setups where islands appear [24]. We find that the bath’s entanglement contour approximately vanishes everywhere except for a finite interval next to the ‘black hole’ degrees of freedom at the end of the half-line, and that this is caused by an island phase transition. This is because the entanglement contour quantifies how well the bath’s state can be reconstructed from its marginals, through its connection to conditional mutual information. Our calculation is only a first step in using local measures of entanglement to probe Hawking radiation, because the setup is static and so not a good model for black hole evaporation. We leave entanglement contours of finite temperature and dynamic black holes to future work."

Report 3 Comment 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.

Clarification added to introduction: "One objection to the formula (1.2) is that mathematically it is a mere repackaging of the entanglement entropy of various subregions, but the same objection can be levelled at quantities generally accepted to be useful such as mutual information and conditional entropy. The value is in the physical interpretation."