Pith. sign in

REVIEW 4 major objections 5 minor 28 references

A revision to the QES prescription

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read The paper argues that the entanglement entropy of an evaporating black hole is a weighted average over all candidate surfaces, not the single extremal surface picked by the QES prescription.

desk verdict A genuine n→1 subtlety in the QES replica derivation, but the proposed weighted-sum fix is postulated rather than derived. read the letter →

arxiv 2506.14071 v1 pith:TFONKL2T submitted 2025-06-17 hep-th

Amir A. Khodahami , Azizollah Azizi This is my paper
classification hep-th
keywords entanglemententropyquantumextremalsurfaceblackholeevaporationreplicatricksaddlepointapproximationgeneralizedislandunitaritycurve
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper argues that the standard quantum extremal surface (QES) prescription for the entanglement entropy of an evaporating black hole is incomplete. In the replica-trick derivation, the term that selects the extremal surface is proportional to $n-1$ and vanishes as the number of replicas $n$ approaches one, so the saddle-point approximation that justifies a single surface breaks down. The authors derive a revised formula in which the entropy is a weighted average over all candidate surfaces $X$ inside the black hole radius, with weight $\exp(-S_{\rm semi\text{-}cl}(\Sigma_X))$, plus a logarithmic term. Early and late in evaporation this formula reproduces the expected unitary behavior, and when the weight is sharply peaked it reduces to the generalized entropy at an extremal surface. If correct, it changes the central object of the calculation from one special surface to an entire distribution of surfaces.

What carries the argument

The central object is a weighted ensemble of codimension-2 surfaces: every surface $X$ between the origin and the black hole radius is assigned weight $\exp(-S_{\rm semi\text{-}cl}(\Sigma_X))$, where $\Sigma_X$ is the region between $X$ and the cutoff surface. The weight is generated by evaluating the on-shell action of the degrees of freedom outside the gluing surface, which carry a factor $n$ in the replicated action; the location of the gluing surface itself is governed by a term of order $n-1$, so no saddle point selects it and the path integral sums over all locations. The same mechanism explains why a naive saddle-point treatment fails: the width of the integrand in the surface location diverges as $n\to 1$ unless a cutoff is introduced, and the divergent derivative of the $n$-dependent replica entropy at $n=1$ forbids a finite cutoff.

What would settle it

Compute the full replicated path integral for a solvable two-dimensional model of an evaporating black hole without replacing the on-shell action of $\Sigma_w$ by its entanglement entropy, and take $n\to 1$; if the result differs from Eq. (4.6), the weighting is wrong.

Watch

Extended reading notes

Core claim

The central claim is Eq. (4.6): the fine-grained entropy of the black hole should be $$S = \frac{\sum_{X\le r_{\rm Sch}} [S_{\rm gen}(X)+S_{\rm semi\text{-}cl}(\Sigma_X)] $e^{{-S_{\rm semi\text{-}}$cl}(\Sigma_X)}}{\sum_{X\le r_{\rm Sch}} $e^{{-S_{\rm semi\text{-}}$cl}(\Sigma_X)}} + \ln \sum_{X\le r_{\rm Sch}} $e^{{-S_{\rm semi\text{-}}$cl}(\Sigma_X)},$$ not the single extremal surface selected by the minimality rule. The weight emerges from the path integral over the region outside each gluing surface, evaluated on shell, and is not visible from the replicated action alone. In the early stages, the rapidly growing semi-classical entropy makes only nearly vanishing surfaces contribute, giving the increasing entropy of the early semi-classical saddle; at late times, only surfaces near the black hole radius contribute, giving the decreasing area term required by unitarity. When the weight is sharply peaked at a particular surface, the formula reduces to the generalized entropy at that surface, recovering the QES answer with a different selection rule.

Load-bearing premise

The calculation assumes, without deriving it from the path integral, that the on-shell action of the region inside each trial surface equals the entanglement entropy of the fields in that region, and if that equality fails the weighted formula collapses.

Editorial extensions

If this is right

  • Early in evaporation the weight is dominated by surfaces near $X=0$, so the entropy grows as the semi-classical entropy of the interior.
  • Late in evaporation only surfaces near $X=r_{\rm Sch}$ contribute, yielding the decreasing area that makes the evolution unitary.
  • When the weight is sharply peaked, the formula reduces to the generalized entropy at an extremal surface, so the QES prescription survives as a special case.
  • Higher replica contributions are expected to change the value of the entropy but leave the weight $\exp(-S_{\rm semi\text{-}cl}(\Sigma_X))$ unchanged.
  • The weight's information-theoretic form implies that the dominant surface maximizes the number of enclosed microstates while minimizing the correlations it cuts across.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If Eq. (4.6) is correct, the QES answer is only the leading term of an expansion; the first corrections are controlled by the spread of $S_{\rm semi\text{-}cl}(\Sigma_X)$ and should be computable in tractable models.
  • A numerical replica calculation in a toy model would test the on-shell identification directly, since the exact integral over gluing-surface locations is in principle well defined.
  • The 'correlations cost' interpretation suggests a variational principle for the island region that could be applied to settings beyond the one considered in the paper.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. This paper revisits the quantum extremal surface (QES) prescription for black hole entanglement entropy. It argues that the standard replica-trick derivation of the QES formula relies on a saddle-point approximation in the number of replicas n that becomes invalid in the limit n → 1, because the prefactor (n−1) suppresses the extremization condition. To address this, the paper proposes a revised entropy formula, Eq. (4.6), which replaces the min-extremum over a single surface by a weighted sum over all surfaces X from 0 to rSch, with weight exp(−S_semi-cl(Σ_X)) plus a logarithmic term. The authors claim that this formula reduces to the generalized entropy at an extremal surface under a sharply-peaked approximation, and that it reproduces the Page curve in early and late evaporation stages.

Significance. If the proposed revision were rigorously derived, it would constitute a significant modification of the QES prescription, altering the island formula and the way extremal surfaces are selected in semiclassical gravity. The paper correctly emphasizes the subtlety of the n → 1 limit in the replica trick, a point that is often glossed over. However, the central new step—replacing the on-shell action of a region by its entanglement entropy—is asserted rather than derived, and the paper itself acknowledges that 'a more detailed analysis was required.' The validation in Section 5 is also circular: it assumes the behavior of S_semi-cl(Σ_X) needed to obtain the Page curve rather than deriving it from an independent calculation. The paper contains no machine-checked proofs or concrete model computations; its contribution is a plausible heuristic proposal, not a supported derivation. For these reasons, the significance of the result is currently outweighed by the unsupported load-bearing step.

major comments (4)
  1. [Section 4, between Eqs. (4.2) and (4.3)] The transition from exp(−n \tilde I(Σ_w)) to exp(−n S(Σ_w)) is the foundation of the entire revised formula, yet it is not justified. The on-shell Euclidean action of a region is a local functional determined by the classical solution, whereas the entanglement entropy S(Σ_w) is a generally non-local functional of the quantum state; there is no general identity equating them. The paper's justification—that the action 'characterizes the degrees of freedom'—is qualitative and does not follow from the replicated path integral presented in Section 4. The authors themselves concede that 'a more detailed analysis was required to demonstrate this.' Since the weight in Eq. (4.6) arises solely from this identification, the central claim is unsupported without this step.
  2. [Section 4, Eq. (4.1)] The decomposition of the replicated path integral into region (i) and region (ii) with the factorized form exp(−(n−1)Sgen(w)) ∫_{region (ii)} exp(−n I) is not derived. The coefficients (n−1) and n are read off from the off-shell action in Eq. (2.3), but it is not shown that the path integral of the full replicated system separates into a product of contributions from the two regions with these weights. In particular, the treatment of the twist operator and the boundary conditions at the gluing surface is not specified, and no concrete model is used to verify the factorization. Without such a derivation, Eq. (4.1) is an assumption about the path integral structure.
  3. [Section 5, Eq. (5.1) and Eq. (5.3)] The validation of the revised formula is circular. At early times, the result S ≈ Sgen(X=0) follows from the assumed linear growth S_semi-cl(Σ_X) ≈ O(N X/rSch), and at late times the result S ≈ Area(X=rSch)/4G follows from the assumed suppression of all X ≠ rSch. These are not independent predictions of Eq. (4.6) but rather properties inserted into S_semi-cl to obtain the Page curve. The paper does not compute S_semi-cl from a specific state or model, nor does it compare the weighted sum against an independent calculation. Consequently, the agreement with the Page curve is an artifact of the assumptions, not evidence for the formula.
  4. [Section 4, Eq. (4.7) and Section 3] The reduction to the generalized entropy at the extremal surface ilde X, Eq. (4.7), assumes that the probability distribution exp(−S_semi-cl(Σ_X)) is sharply peaked. However, Section 3 argues that the analogous distribution exp(−(n−1)Sgen(w)) is not sharply peaked because the prefactor (n−1) vanishes as n → 1. The paper does not explain why the peak in exp(−S_semi-cl) is justified when no small parameter multiplies S_semi-cl in the exponent; the distinction is not self-evident and requires a separate argument beyond the heuristic that S_semi-cl is large.
minor comments (5)
  1. [Section 3, Eq. (1.4)] The expression for the derivative of the Rényi entropy near n = 1 is written as d/dn Sn|_{n→1} = S/(2(n−1))|_{n→1} → ∞, which is ambiguous: the divergence is not literally S/(2(n−1)) but arises from the second-order expansion of Tr ρ^n; the notation should be clarified.
  2. [Section 5, first sentence] The phrase 'Let check whether' should be 'Let us check whether'.
  3. [Section 2, Fig. 2 caption] The caption reads 'Left is what Hawking considered in his calculations' which is informal; it should say 'The left panel shows the Hawking saddle'.
  4. [Section 4, Eqs. (4.4)–(4.5)] The subscript w in ⟨Sgen(w)⟩_w and ⟨S(Σ_w)⟩_w is reused for the summation variable, which can be confusing; using a different index (e.g., j) for the expectation value would improve readability.
  5. [References] The papers [18,19] contain extensive discussion of the n → 1 limit and the factorization of the replica path integral; the manuscript would benefit from engaging with those explicit analyses rather than only citing them as the source of the off-shell action.

Circularity Check

2 steps flagged · score 7.0 of 10

The weight in the revised entropy formula is set equal to exp(-S(Σ_w)) by an asserted action-entropy identification, and the Section 5 'validation' then feeds assumed profiles of the same S into that weight; the resulting Page-curve peaks are forced by construction.

  1. self definitional [Section 4, between Eqs. (4.2) and (4.3)]
    "This action characterizes the degrees of freedom contained in Σw, allowing us to replace it with the entanglement entropy of the fields on Σw, i.e., S(Σw); hence tr(ρn) = Σw exp(−(n−1)Sgen(w)) exp(−nS(Σw))."

    The load-bearing input of the revised formula is obtained by replacing the on-shell action Ĩ(Σw) with the entanglement entropy S(Σw) purely by assertion ('characterizes the degrees of freedom'), not by any path-integral identity. The weight exp(−S(Σw)) in Eq. (4.3), and hence the final weight exp(−Ssemi-cl(ΣX)) in Eq. (4.6), is therefore postulated rather than derived. Any later statement about where this weight is peaked is a property of the postulated input, not an independent prediction. The paper itself flags the gap: 'a more detailed analysis was required to demonstrate this, going beyond general arguments based solely on the action.'

  2. fitted input called prediction [Section 5, Eqs. (5.1)–(5.3)]
    "Ssemi-cl(ΣX) ≈ O(N X / rSch). Consequently, exp(−Ssemi-cl(ΣX)) decreases at a very fast rate, resulting in substantial contributions ... only from surfaces that are nearly vanishing (X ≈ 0). So we obtain S(t ≪ thalf) ≈ Sgen(X = 0) ... only when X ≈ rSch we see a non-negligible contribution ... Hence Eq. (4.7) gives S(t ≫ thalf) ≈ Sgen(X = rSch) ≈ Area(X = rSch)/4G, which is the desired replica wormhole."

    The 'validation' inserts assumed forms of Ssemi-cl(ΣX) into the very weight exp(−Ssemi-cl(ΣX)) that was defined by the same quantity in Eq. (4.3). The early-time peak at X≈0 follows from assuming Ssemi-cl grows with X; the late-time peak at X≈rSch follows from assuming Ssemi-cl is small only at rSch. No independent observable is computed; the Page curve is recovered because the assumed entropy profile was placed in the exponent. The output is determined by the input function, so calling this a validation is circular.

full rationale

Most of the paper (Sections 2–3) is a review of the standard QES derivation and the n→1 saddle-point problem, which is not circular. The circularity enters at the one load-bearing step in Section 4: Eq. (4.2) contains exp(−nĨ(Σw)) and Eq. (4.3) replaces this by exp(−nS(Σw)) on the assertion that the on-shell action 'characterizes the degrees of freedom' in Σw. No derivation from the replicated path integral is supplied; the paper concedes that 'a more detailed analysis was required to demonstrate this.' The final formula (4.6) therefore has the weight exp(−Ssemi-cl(ΣX)) built in by hand. The Section 5 'validation' then substitutes assumed behaviors of Ssemi-cl(ΣX) (growing linearly with X early, large everywhere except at rSch late) into that same weight; the resulting peaks at X=0 and X=rSch are forced by those assumptions. Thus the Page-curve check is a consistency check of the ansatz, not an independent confirmation. There is no load-bearing self-citation chain or uniqueness import; the central issue is that the defining identification (action → entropy) is postulated, and the 'predictions' of the peaks are algebraic consequences of the postulated weight.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no fitted numerical parameters, but the weight exp(-S(Sigma_X)) is an unproven functional choice. The derivation relies on several ad hoc decompositions and identifications that are not backed by a first-principles path integral computation.

assumptions (5)
  • ad hoc to paper The replicated action can be decomposed into a region inside the gluing surface with weight (n-1) Sgen(w) and a region outside with weight n I.
    Stated in Section 4 before Eq (4.1), but not derived from the path integral. This decomposition is essential for obtaining the weighted sum.
  • ad hoc to paper The on-shell action of the region Sigma_w can be replaced by the entanglement entropy S(Sigma_w) of the quantum fields on that region.
    This identification, between Eqs (4.2) and (4.3), is the key step producing the weight exp(-S(Sigma_w)). No justification is given.
  • domain assumption The replica trick with analytic continuation to n=1 is valid despite the non-differentiability of the Rényi entropy at n=1.
    The paper criticizes the use of a cutoff in n, but still uses the replica formula to define the entropy, without resolving the non-differentiability issue.
  • ad hoc to paper Only surfaces X with 0 <= X <= rSch contribute to the sum.
    Imposed in Eq (4.6) with no derivation; surfaces outside the horizon are excluded without discussion.
  • domain assumption The probability distribution is sharply peaked at the surface that extremizes S_semi-cl(Sigma_X).
    Needed to reduce Eq (4.6) to the QES form; asserted in Section 4 without proof.

how reviews work

0 comments
Cite this review

Pith. "Pith review of A revision to the QES prescription." pith.science (2026). https://pith.science/paper/TFONKL2T

@misc{pith2026250614071,
  author       = {Pith},
  title        = {Pith review of: A revision to the QES prescription},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TFONKL2T}},
  note         = {Machine review of arXiv:2506.14071}
}
read the original abstract

In this paper, we propose a revision to the Quantum Extremal Surface (QES) prescription, which plays a crucial role in describing the entanglement entropy of black holes. While derivations exist for the original QES prescription using the replica trick, they rely on a saddle point approximation that faces challenges when the number of replicas approaches unity. We highlight these challenges and derive a refined formula for the entanglement entropy that incorporates a weighted summation over multiple surfaces, contrasting the QES prescription's reliance on a single extremal surface. We validate our formulation by demonstrating consistency with expected results for both the early and late stages of black hole evaporation, confirming its alignment with the unitary evolution depicted by the Page curve.

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

28 extracted references · 2 canonical work pages

  1. [1]

    Hawking, Breakdown of predictability in gravitational collapse , Phys

    S.W. Hawking, Breakdown of predictability in gravitational collapse , Phys. Rev. D 14 (1976) 2460

    1976

  2. [2]

    Ryu and T

    S. Ryu and T. Takayanagi, Holographic Derivation of Entanglement Entropy from AdS/CFT , Phys. Rev. Lett. 96 (2006) 181602 [ hep-th/0603001]

    arXiv 2006

  3. [3]

    Fursaev, Proof of the Holographic Formula for Entanglement Entropy , J

    D.V. Fursaev, Proof of the Holographic Formula for Entanglement Entropy , J. High Energy Phys. 2006 (2006) 018 [ hep-th/0606184]

    arXiv 2006

  4. [4]

    Nishioka, S

    T. Nishioka, S. Ryu and T. Takayanagi, Holographic Entanglement Entropy: An Overview , J. Phys. A: Math. Theor. 42 (2009) 504008 [ 0905.0932]

    arXiv 2009

  5. [5]

    Hubeny, M

    V.E. Hubeny, M. Rangamani and T. Takayanagi, A Covariant Holographic Entanglement Entropy Proposal, J. High Energy Phys. 2007 (2007) 062 [ 0705.0016]

    arXiv 2007

  6. [6]

    Faulkner, A

    T. Faulkner, A. Lewkowycz and J. Maldacena, Quantum corrections to holographic entanglement entropy, J. High Energ. Phys. 2013 (2013) 74 [ 1307.2892]

    arXiv 2013

  7. [7]

    Engelhardt and A.C

    N. Engelhardt and A.C. Wall, Quantum Extremal Surfaces: Holographic Entanglement Entropy beyond the Classical Regime , J. High Energ. Phys. 2015 (2015) 73 [ 1408.3203]

    arXiv 2015

  8. [8]

    Almheiri, T

    A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian and A. Tajdini, The entropy of Hawking radiation, Rev. Mod. Phys. 93 (2021) 035002 [ 2006.06872]

    arXiv 2021

Show all 28 references
  1. [9]

    Entropy Evaporated by a Black Hole

    D.N. Page, Comment on “Entropy Evaporated by a Black Hole” , Phys. Rev. Lett. 50 (1983) 1013

    1983

  2. [10]

    Page, Average Entropy of a Subsystem , Phys

    D.N. Page, Average Entropy of a Subsystem , Phys. Rev. Lett. 71 (1993) 1291 [gr-qc/9305007]

    1993 arXiv

  3. [11]

    Page, Time Dependence of Hawking Radiation Entropy , J

    D.N. Page, Time Dependence of Hawking Radiation Entropy , J. Cosmol. Astropart. Phys. 2013 (2013) 028 [ 1301.4995]. – 10 –

    2013 arXiv

  4. [12]

    Gautason, L

    F.F. Gautason, L. Schneiderbauer, W. Sybesma and L. Thorlacius, Page Curve for an Evaporating Black Hole , J. High Energ. Phys. 2020 (2020) 91 [ 2004.00598]

    2020 arXiv

  5. [13]

    Hartman, E

    T. Hartman, E. Shaghoulian and A. Strominger, Islands in Asymptotically Flat 2D Gravity , J. High Energ. Phys. 2020 (2020) 22 [ 2004.13857]

    2020 arXiv

  6. [14]

    X. Wang, R. Li and J. Wang, Page curves for a family of exactly solvable evaporating black holes, Phys. Rev. D 103 (2021) 126026 [ 2104.00224]

    2021 arXiv

  7. [15]

    Harlow, Jerusalem Lectures on Black Holes and Quantum Information , Rev

    D. Harlow, Jerusalem Lectures on Black Holes and Quantum Information , Rev. Mod. Phys. 88 (2016) 015002 [ 1409.1231]

    2016 arXiv

  8. [16]

    Witten, Introduction to Black Hole Thermodynamics [2412.16795]

    E. Witten, Introduction to Black Hole Thermodynamics [2412.16795]

    arXiv

  9. [17]

    Headrick, Entanglement Renyi entropies in holographic theories , Phys

    M. Headrick, Entanglement Renyi entropies in holographic theories , Phys. Rev. D 82 (2010) 126010 [1006.0047]

    2010 arXiv

  10. [18]

    Almheiri, T

    A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian and A. Tajdini, Replica Wormholes and the Entropy of Hawking Radiation , J. High Energ. Phys. 2020 (2020) 13 [ 1911.12333]

    2020 arXiv

  11. [19]

    Penington, S.H

    G. Penington, S.H. Shenker, D. Stanford and Z. Yang, Replica wormholes and the black hole interior, J. High Energ. Phys. 2022 (2022) 205 [ 1911.11977]

    2022 arXiv

  12. [20]

    Mertens and G.J

    T.G. Mertens and G.J. Turiaci, Solvable Models of Quantum Black Holes: A Review on Jackiw-Teitelboim Gravity, Living Rev Relativ 26 (2023) 4 [ 2210.10846]

    2023 arXiv

  13. [21]

    Lewkowycz and J

    A. Lewkowycz and J. Maldacena, Generalized gravitational entropy, J. High Energ. Phys. 2013 (2013) 90 [ 1304.4926]

    2013 arXiv

  14. [22]

    Engelhardt and H

    N. Engelhardt and H. Liu, Algebraic ER=EPR and Complexity Transfer , J. High Energ. Phys. 2024 (2024) 13 [ 2311.04281]

    2024 arXiv

  15. [23]

    Engelhardt and ˚A

    N. Engelhardt and ˚A. Folkestad, Canonical Purification of Evaporating Black Holes , Phys. Rev. D 105 (2022) 086010 [ 2201.08395]

    2022 arXiv

  16. [24]

    Maldacena and L

    J. Maldacena and L. Susskind, Cool horizons for entangled black holes , Fortschritte der Physik 61 (2013) 781 [ 1306.0533]

    2013 arXiv

  17. [25]

    Antonini, C.-H

    S. Antonini, C.-H. Chen, H. Maxfield and G. Penington, An apologia for islands [2506.04311]

    arXiv

  18. [26]

    Raamsdonk, Building up spacetime with quantum entanglement , Gen Relativ Gravit 42 (2010) 2323 [ 1005.3035]

    M.V. Raamsdonk, Building up spacetime with quantum entanglement , Gen Relativ Gravit 42 (2010) 2323 [ 1005.3035]

    2010 arXiv

  19. [27]

    Raamsdonk, Comments on quantum gravity and entanglement [0907.2939]

    M.V. Raamsdonk, Comments on quantum gravity and entanglement [0907.2939]

    arXiv

  20. [28]

    Blommaert, J

    A. Blommaert, J. Kruthoff and S. Yao, The power of Lorentzian wormholes , J. High Energ. Phys. 2023 (2023) 5. – 11 –

    2023

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.