Puzzles and pitfalls involving Haar-typicality in holography

We thank the referees for their reports. Based on their comments and suggestions, we have prepared a revised draft at the request of the editor-in-charge. The revised draft is v5 on the arXiv. Changes, sorted according to referee, are listed below.

Report 1:

• The changes described in our reply and that we made in v4 were brought forward to v5. The sentence that we added in the introduction was moved to a footnote on p.3, and the new section 5.3 remains in-place.

Report 2:

• We added the heuristic example to section 5.3. We did not include our detailed rebuttal in the draft, as it is publicly-visible on this submission page.

Report 3:

• We amended the text throughout to remove any use of the term "Haar-typical state of a CFT". (This includes a revision to the abstract.) We agree with the referee that the term is ill-defined---Haar-typicality indeed requires a finite-dimensional Hilbert space.

• We further highlight this point in the 6th paragraph of the introduction, and we cleaned up the rest of this paragraph accordingly.

• Likewise, we re-wrote section 2 starting at the last paragraph on p.4 to be more careful about the finite-dimensional Hilbert space that is to be used in place of the full continuum CFT for defining Haar-typicality. Rather than focus on a specific lattice model, we elected to write down what properties we assume for a finite-dimensional model and we give several examples of models with these properties in the literature, including the critical spin chain.

• We acknowledged the subtlety that the referee raised regarding the relationship between UV cutoffs in the bulk and boundary in a footnote on p.5.

Further comments:

• We made small aesthetic formatting changes and corrected minor typos throughout.

• We also added several references.