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Puzzles and pitfalls involving Haar-typicality in holography

by Ning Bao, Aidan Chatwin-Davies

This Submission thread is now published as SciPost Phys. 4, 033 (2018)

Submission summary

As Contributors: Aidan Chatwin-Davies
Arxiv Link: (pdf)
Date accepted: 2018-06-08
Date submitted: 2018-05-25 02:00
Submitted by: Chatwin-Davies, Aidan
Submitted to: SciPost Physics
Academic field: Physics
  • Gravitation, Cosmology and Astroparticle Physics
  • High-Energy Physics - Theory
  • Quantum Physics
Approach: Theoretical


Holographic states that have a well-defined geometric dual in AdS/CFT are not faithfully represented by Haar-typical states in finite-dimensional models. As such, trying to apply principles and lessons from Haar-random ensembles of states to holographic states can lead to apparent puzzles and contradictions. We point out a handful of these pitfalls.

Ontology / Topics

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AdS/CFT correspondence Haar typicality Holography

Published as SciPost Phys. 4, 033 (2018)

Author comments upon resubmission

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.

List of changes

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.

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