Large Breakdowns of Entanglement Wedge Reconstruction

Adam Levine; Chris Akers; Stefan Leichenauer

REVIEW 3 major objections 4 minor 2 cited by

Large Breakdowns of Entanglement Wedge Reconstruction

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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

Pith's one-line read The bulk reconstructable from a boundary subregion can be arbitrarily smaller than its entanglement wedge, even with small backreaction and no horizon.

desk verdict Trust the dustball, not the near-vacuum qubits — the reconstruction wedge concept is solid and the paper deserves refereeing, but Section II's O(1) entropy differences are too fragile. read the letter →

arxiv 1908.03975 v1 pith:G2ONAQ7S submitted 2019-08-12 hep-th

Chris Akers , Stefan Leichenauer , Adam Levine This is my paper · ORCID

classification hep-th

keywords AdS/CFTentanglementwedgereconstructionquantumextremalsurfaceerrorcorrectionstate-dependentdustballholographicentropy

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 tries to establish that in AdS/CFT the bulk region reconstructable from a boundary subregion A — the reconstruction wedge, defined as the intersection of A's entanglement wedges over all states in a code subspace — can be arbitrarily smaller than the entanglement wedge of a particular pure state, even when backreaction is small and the setting is far from black hole interiors. It constructs code subspaces whose states share one bulk geometry but differ in internal spin degrees of freedom. Near an entanglement phase transition in the vacuum, a handful of qubits can exclude the 'middle' of the bulk from the reconstruction wedge; in a low-density dustball with entropy of order $1/G_N$, the reconstruction wedge excludes the whole dustball while the pure-state entanglement wedge contains it. The result implies that bulk reconstruction is generically subspace-dependent, not a special feature of black hole physics.

What carries the argument

The central object is the reconstruction wedge, the intersection of all entanglement wedges of A over states in the code subspace. The mechanism that does the work is the competition between bulk entropy and surface area in the generalized entropy: because the quantum extremal surface is selected by minimizing generalized entropy, a bulk state with high entropy can shift the surface and exclude a region that lower-entropy states would include. Bell pairs act as a controlled entropy budget that can be spent to keep the middle inside the reconstruction wedge, while the dustball supplies an entropy reservoir of order $1/G_N$ without changing the metric at the order of interest.

What would settle it

Perform a detailed calculation of the generalized entropy of the two candidate quantum extremal surfaces in the four-interval AdS$_3$ vacuum with one mixed qubit in the middle, including graviton fluctuations; if the fluctuations wash out the $O(1)$ entropy difference, the middle remains reconstructable and the near-vacuum separation disappears. For the dustball, compute the quantum extremal surface of A for a pure dustball state within the code subspace that also contains the maximally mixed state; if the surface does not move by an $O(1/G_N)$ area as the state changes, the claimed macroscopic gap is absent.

Watch

Extended reading notes

Core claim

The central claim is that the reconstruction wedge — the intersection of the entanglement wedges of A for every pure or mixed state in a code subspace — can be macroscopically smaller than the entanglement wedge of any given pure state. The paper proves this by explicit construction in AdS$_3$. With the boundary split into four equal intervals and a single qubit at the center, the maximally mixed state of the qubit breaks the degeneracy between the two competing extremal surfaces and pushes the entanglement wedge of A away from the middle, so the reconstruction wedge of A excludes it. Adding Bell pairs can restore the middle to A's reconstruction wedge, one Bell pair per qubit, showing a precise counting rule. In the dustball example, a spherical cloud of pressureless dust with total entropy of order $1/G_N$ sits at the center; a boundary region A formed from many small intervals has a quantum extremal surface whose area is only $O(G_N^0)$ away from its complement's, so the dustball's entropy dominates and excludes the dustball from the reconstruction wedge even though pure states include it. These examples also exhibit quantum extremal surfaces macroscopically different from the classical RT surface.

Load-bearing premise

The weakest load-bearing premise is that an $O(1)$ entropy difference can robustly select the quantum extremal surface and fix the reconstruction wedge, despite graviton-induced area fluctuations of order $\sqrt{G_N}$ that are much larger than that entropy difference in the classical limit.

Editorial extensions

If this is right

Complementary recovery — the statement that A and its complement reconstruct complementary bulk regions — fails macroscopically whenever the code subspace contains a sufficiently entropic mixed state.
Entanglement wedge reconstruction as usually stated holds only for fixed code subspaces; a single reconstruction prescription cannot work on the full code subspace when the reconstruction wedge is smaller.
The quantum extremal surface can jump from the near-A surface to the near-complement surface as the bulk state varies within one code subspace, so the location of the QES is genuinely state-dependent even in near-vacuum geometries without horizons.
Boundary regions composed of $O(G_N)$-sized intervals have severely limited reconstructability, since that is the regime where the dustball example shows the largest separation.

Reading between the lines

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

If the $O(1)$-entropy caveat is overcome with fixed-area states, the qubit examples would imply that even a single mixed bulk qubit can dictate the reconstruction wedge; the threshold for state-dependence would be entropy of order one, not order $1/G_N$.
A natural boundary diagnostic suggested by the dustball construction is a sharp drop in the mutual information between A and the dustball degrees of freedom as the bulk state passes from pure to maximally mixed, which could be probed holographically.
The same intersection construction could be applied to other holographic settings, such as evaporating black holes, where the reconstruction wedge may track the island region, suggesting a common mechanism: the most entropic state in the code subspace controls the reconstructable region.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague,

The thing to know: this paper gives us a useful, explicit notion of the reconstruction wedge and a convincing dustball example where it is macroscopically smaller than the entanglement wedge with no horizon in sight. The near-vacuum examples in Section II are the advertised headline, and they are the weak part; the O(1) entropy differences at the phase transition are swamped by graviton area fluctuations, as the authors admit in footnote 6. So treat the near-vacuum claim as not established.

What's new: Hayden-Penington had state-dependence for black holes; this paper decouples the phenomenon from horizons. The reconstruction wedge definition—intersection of entanglement wedges over all states in the code subspace, including mixed—is clean and should stick. The dustball code subspace has O(1/G_N) entropy, so the separation between RW and EW can be made arbitrarily large while backreaction stays small. That example is the real contribution and it holds up. The observation that QES can be macroscopically different from RT is also useful.

Soft spots: Section II's whole mechanism is a degeneracy between RT surfaces broken by one qubit's entropy. The generalized-entropy difference is O(1), but the area fluctuations are order sqrt(l/G_N), which diverges as G_N -> 0. The authors' footnote 6 is honest but it undercuts the section. The Bell-pair bookkeeping is cute, but it doesn't fix the fluctuation problem. For the dustball, the QES positions are asserted rather than derived; I believe the claims are right—the low-density, large-radius limit makes it plausible—but a referee should ask for the computation or at least a sharper estimate. The higher-dimensional generalization in Section IV is a sketch, not an argument. Citation pattern is fine; they build on HP and Engelhardt-Wall appropriately.

Bottom line: the paper deserves a serious referee. The dustball construction alone is worth publishing. A revision should either put the near-vacuum examples in fixed-area states or clearly demote them to heuristic illustrations. I'd bring it to the reading group and would cite it.

Recommendation: send to peer review, with the expectation of a major revision on Section II.

Referee Report

3 major / 4 minor

Summary. The paper defines the reconstruction wedge of a boundary region A as the intersection, over all states in a code subspace, of the entanglement wedges of A. It then constructs two families of examples in AdS3/CFT2 where this reconstruction wedge is claimed to be macroscopically smaller than the entanglement wedge of a particular state. The first family (Section II) places a few bulk qubits and Bell pairs at an entanglement phase transition in near-vacuum states. The second (Section III) considers a large, low-density dustball with O(1/G_N) internal entropy and a boundary region A made of O(G_N^{-1}) small intervals. The paper argues that in the maximally mixed dustball state, the entanglement wedges of A and its complement exclude the dustball, while in a pure dustball state A's entanglement wedge includes it, leading to an arbitrarily large gap.

Significance. If established, the result would show that subspace-dependent bulk reconstruction is not a peculiarity of black hole horizons, and that quantum extremal surfaces can differ macroscopically from classical RT surfaces. The paper is clearly written and the dustball construction is a useful tractable model. The definition of the reconstruction wedge is clean, and the paper correctly emphasizes the role of mixed states in the code subspace. The dustball example, which uses O(1/G_N) entropy differences, appears to avoid the main caveat affecting the near-vacuum examples. However, the near-vacuum claims are not presently supported, and the dustball geometry requires explicit checks.

major comments (3)

[II, footnote 6] The near-vacuum examples select a unique QES using O(1) differences in generalized entropy at an exact phase transition. As footnote 6 concedes, graviton area fluctuations give delta A/(4G_N) ~ sqrt(l_AdS/G_N), which diverges in the G_N -> 0 limit and is much larger than the O(1) entropy differences. Therefore the QES prescription is not controlled for these code subspaces, and the abstract's claim of arbitrarily large separations in near-vacuum states is not established. The dustball example is not subject to this objection, but the Section II examples and the abstract need to be revised, for example by using fixed-area states as the footnote suggests.
[III.C] The central geometric input is that for n ~ R/l_AdS the QES of every connected component of A and \bar A lies outside the dustball, and that the area difference between the two candidate configurations is Eq. (9) with an O(1) constant gamma independent of n. These facts are asserted without derivation. An explicit computation using the metrics in Eqs. (3)-(8) is needed to verify the scaling and to confirm that the dustball entropy can dominate the area difference while the QES remain outside the dustball.
[II and Definition in Introduction] The reconstruction wedge is defined as the intersection over all states in the code subspace, but the examples include states in which the QES is exactly degenerate. For instance, in the Bell Pair 1 example, the maximally mixed state of Qubit 1 gives equal generalized entropy for including and excluding the middle, so the entanglement wedge is not unique. The paper states that the EW includes the middle for all states in this code subspace, but under a strict intersection over all degenerate choices the middle would be excluded for that state. A tie-breaking prescription or a fixed-area-state treatment is required before the Section II conclusions follow.

minor comments (4)

[II] In the paragraph after Figure 1, 'reigon' should be 'region'.
[III.A] After Eq. (2), the text should state explicitly that eta is the small parameter controlling the validity of the near-vacuum exterior metric in Eq. (8), and that all subsequent equations are leading-order in eta.
[IV] The extension to even dimensions is presented as an expectation; it would be clearer to label it as a conjecture and indicate what a proof would require.
[Footnote 3] The claim that dressing the qubit equally to A and \bar A 'does not affect the degeneracy' deserves a fuller explanation, since the dressing crosses an RT surface and changes its area by an amount of order the qubit entropy.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the paper's examples are direct constructions using the standard quantum extremal surface prescription and the Hayden-Penington definition of the reconstruction wedge.

full rationale

The paper's derivation chain is not circular. It defines the reconstruction wedge, following Hayden and Penington, as the intersection of the entanglement wedges of a boundary region over all states in a code subspace, and then explicitly computes entanglement wedges using the standard quantum extremal surface prescription of Engelhardt and Wall. The central claim—that the reconstruction wedge can be macroscopically smaller than the entanglement wedge—is a constructed consequence of these definitions and the chosen states, not a quantity fitted from the data it purports to predict. No parameter is tuned to enforce the result, and no 'prediction' is equivalent to an input by construction. The only self-citation, reference [13] in footnote 6, is a passing suggestion about fixed-area states as a possible mitigation and is not load-bearing. The explicit caveat in footnote 6, that O(1) generalized-entropy differences in the near-vacuum examples may be swamped by graviton area fluctuations of order sqrt(l/G_N), is a limitation of those examples rather than a circular step; the paper honestly flags that it is 'unclear whether A can really reconstruct the middle.' The dustball example, with entropy O(1/G_N), does not rely on this fragile O(1) comparison and still supports the broader claim. Thus the derivation is self-contained against standard external inputs and contains no circular reduction.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The central claim does not introduce new physical entities or fitted parameters; it is built on standard holographic assumptions (QES prescription, relative entropy equality, HP reconstruction wedge) plus the paper's specific idealizations: degenerate spin states with no backreaction, the dustball geometry, and the assumed placement of QESs. The key fragility is the O(1) entropy selection of QES discussed in footnote 6.

free parameters (4)

eta
Dimensionless energy density parameter eta = 8*pi*G_N*rho0*l_AdS^2, chosen small to keep backreaction under control; not fitted to data.
R
Dustball radius at t=0, chosen large enough (with n ~ R/l_AdS) so the dustball entropy is O(G_N^{-1}) and dominates the area difference Delta A; a free choice in the construction.
n
Number of boundary intervals (2n+1), chosen ~ R/l_AdS ~ O(G_N^{-1}) to keep QESs outside the dustball; a free parameter of the construction.
k
Number of qubits in the middle (near-vacuum example) or number of dust particles (dustball), chosen to satisfy k log 2 < Delta A/(4 G_N) or k = O(G_N^{-1}); dimension of the code subspace is a free choice.

assumptions (6)

domain assumption The quantum extremal surface prescription gives the boundary entanglement entropy and defines the entanglement wedge at all orders in G_N.
Used throughout (Section I and II) as the basis for computing entanglement wedges; standard in holography.
domain assumption Bulk and boundary relative entropies agree to leading order in G_N, S(rho_A||sigma_A) = S(rho_a||sigma_a) + O(G_N), which underpins entanglement wedge reconstruction.
Equation (1), Section I, cited from Jafferis-Lewkowycz-Maldacena-Suh; used as input for reconstruction.
domain assumption The reconstruction wedge is the intersection of entanglement wedges of A over all states in the code subspace, including mixed states.
Section I definition; draws on Hayden-Penington. The paper adopts this as the operative definition without proof.
ad hoc to paper For the dustball code subspace, all internal spin states of the dust particles are exactly degenerate in energy and do not backreact on the geometry, so all states share the same metric.
Section III.B; a simplifying idealization that makes the construction tractable. If spin-state energy differences were non-negligible, the identical-geometry assumption would fail.
domain assumption The dustball interior is described by a portion of FRW and the exterior by BTZ, with metric matching at r=R; the Oppenheimer-Snyder model is valid in 2+1d AdS.
Section III.A; the geometry is assumed as a model of a large, low-density dust cloud.
ad hoc to paper With n ~ R/l_AdS, the quantum extremal surfaces for A and \bar A lie entirely outside the dustball, so the bulk entropy of the dustball is in the region between the candidate surfaces.
Section III.C; the paper argues this but does not provide an explicit computation; the dustball result depends on it.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Large Breakdowns of Entanglement Wedge Reconstruction." pith.science (2026). https://pith.science/paper/G2ONAQ7S

@misc{pith2026190803975,
  author       = {Pith},
  title        = {Pith review of: Large Breakdowns of Entanglement Wedge Reconstruction},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/G2ONAQ7S}},
  note         = {Machine review of arXiv:1908.03975}
}

read the original abstract

We show that the bulk region reconstructable from a given boundary subregion --- which we term the reconstruction wedge --- can be much smaller than the entanglement wedge even when backreaction is small. We find arbitrarily large separations between the reconstruction and entanglement wedges in near-vacuum states for regions close to an entanglement phase transition, and for more general regions in states with large energy (but very low energy density). Our examples also illustrate situations for which the quantum extremal surface is macroscopically different from the Ryu-Takayanagi surface.

Figures

Figures reproduced from arXiv: 1908.03975 by the authors.

**Figure 2.** FIG. 2: In blue is the reconstruction wedge of [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** FIG. 3: The Penrose diagram of the [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: FIG. 4: In blue is the [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Holography at Finite N: Breakdown of Bulk Reconstruction for Subregions
hep-th 2025-08 conditional novelty 6.0 of 10

Reconstructed AdS-Rindler bulk operators have three-point functions scaling as N^{-1} exp((pi/2)(|lambda|-omega)), so they break down above a (2/pi) ln N momentum scale.
An apologia for islands
hep-th 2025-06 conditional novelty 6.0 of 10

Entanglement islands and Page curves can arise in massless gravity without an external bath, and compactly supported gauge-invariant operators exist in islands around generic symmetry-breaking black hole backgrounds.

Reference graph

Works this paper leans on

14 extracted references · 1 canonical work pages · cited by 2 Pith papers

[1]

Bulk Locality and Quantum Error Correction in AdS/CFT,

A. Almheiri, X. Dong, and D. Harlow, “Bulk Locality and Quantum Error Correction in AdS/CFT,” JHEP 04 (2015) 163, arXiv:1411.7041 [hep-th]

arXiv 2015
[2]

Reconstruction of Bulk Operators Within the Entanglement Wedge in Gauge-Gravity Duality,

X. Dong, D. Harlow, and A. C. Wall, “Reconstruction of Bulk Operators Within the Entanglement Wedge in Gauge-Gravity Duality,” Phys. Rev. Lett. 117 no. 2, (2016) 021601, arXiv:1601.05416 [hep-th]

arXiv 2016
[3]

The Ryu-Takayanagi Formula from Quantum Error Correction,

D. Harlow, “The Ryu-Takayanagi Formula from Quantum Error Correction,” arXiv:1607.03901 [hep-th]

arXiv
[4]

Aspects of Holographic Entanglement Entropy,

S. Ryu and T. Takayanagi, “Aspects of Holographic Entanglement Entropy,” JHEP 08 (2006) 045, arXiv:hep-th/0605073 [hep-th]

arXiv 2006
[5]

Holographic derivation of entanglement entropy from AdS/CFT,

S. Ryu and T. Takayanagi, “Holographic derivation of entanglement entropy from AdS/CFT,” Phys. Rev. Lett. 96 (2006) 181602, arXiv:hep-th/0603001 [hep-th]

arXiv 2006
[6]

Relative entropy equals bulk relative entropy,

D. L. Jaﬀeris, A. Lewkowycz, J. Maldacena, and S. J. Suh, “Relative entropy equals bulk relative entropy,” JHEP 06 (2016) 004, arXiv:1512.06431 [hep-th]

arXiv 2016
[7]

Quantum Extremal Surfaces: Holographic Entanglement Entropy Beyond the Classical Regime,

N. Engelhardt and A. C. Wall, “Quantum Extremal Surfaces: Holographic Entanglement Entropy Beyond the Classical Regime,” JHEP 01 (2015) 073, arXiv:1408.3203 [hep-th]

arXiv 2015
[8]

state-dependent

demonstrated that for large enough code subspaces this intuition is false. Consider trying to reconstruct a black hole operator using a boundary region A that is barely more than half the CFT. If this black hole were in a pure state, the entanglement wedge of A would include the black hole. However, if our code subspace included, say, all eSBH black hole ...

work page

Show all 14 references

[9]

Learning the Alpha-bits of Black Holes,

P. Hayden and G. Penington, “Learning the Alpha-bits of Black Holes,” arXiv:1807.06041 [hep-th]

arXiv
[10]

An Infalling Observer in AdS/CFT,

K. Papadodimas and S. Raju, “An Infalling Observer in AdS/CFT,” JHEP 10 (2013) 212, arXiv:1211.6767 [hep-th]

2013 arXiv
[11]

Entanglement Wedge Reconstruction and the Information Paradox,

G. Penington, “Entanglement Wedge Reconstruction and the Information Paradox,” arXiv:1905.08255 [hep-th]

1905 arXiv
[12]

The entropy of bulk quantum ﬁelds and the entanglement wedge of an evaporating black hole,

A. Almheiri, N. Engelhardt, D. Marolf, and H. Maxﬁeld, “The entropy of bulk quantum ﬁelds and the entanglement wedge of an evaporating black hole,” arXiv:1905.08762 [hep-th]

1905 arXiv
[13]

Flat entanglement spectra in ﬁxed-area states of quantum gravity,

X. Dong, D. Harlow, and D. Marolf, “Flat entanglement spectra in ﬁxed-area states of quantum gravity,” arXiv:1811.05382 [hep-th]

arXiv
[14]

Holographic Renyi Entropy from Quantum Error Correction,

C. Akers and P. Rath, “Holographic Renyi Entropy from Quantum Error Correction,” JHEP 05 (2019) 052, arXiv:1811.05171 [hep-th]

2019 arXiv

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Bulk Locality and Quantum Error Correction in AdS/CFT,

A. Almheiri, X. Dong, and D. Harlow, “Bulk Locality and Quantum Error Correction in AdS/CFT,” JHEP 04 (2015) 163, arXiv:1411.7041 [hep-th]

arXiv 2015

[2] [2]

Reconstruction of Bulk Operators Within the Entanglement Wedge in Gauge-Gravity Duality,

X. Dong, D. Harlow, and A. C. Wall, “Reconstruction of Bulk Operators Within the Entanglement Wedge in Gauge-Gravity Duality,” Phys. Rev. Lett. 117 no. 2, (2016) 021601, arXiv:1601.05416 [hep-th]

arXiv 2016

[3] [3]

The Ryu-Takayanagi Formula from Quantum Error Correction,

D. Harlow, “The Ryu-Takayanagi Formula from Quantum Error Correction,” arXiv:1607.03901 [hep-th]

arXiv

[4] [4]

Aspects of Holographic Entanglement Entropy,

S. Ryu and T. Takayanagi, “Aspects of Holographic Entanglement Entropy,” JHEP 08 (2006) 045, arXiv:hep-th/0605073 [hep-th]

arXiv 2006

[5] [5]

Holographic derivation of entanglement entropy from AdS/CFT,

S. Ryu and T. Takayanagi, “Holographic derivation of entanglement entropy from AdS/CFT,” Phys. Rev. Lett. 96 (2006) 181602, arXiv:hep-th/0603001 [hep-th]

arXiv 2006

[6] [6]

Relative entropy equals bulk relative entropy,

D. L. Jaﬀeris, A. Lewkowycz, J. Maldacena, and S. J. Suh, “Relative entropy equals bulk relative entropy,” JHEP 06 (2016) 004, arXiv:1512.06431 [hep-th]

arXiv 2016

[7] [7]

Quantum Extremal Surfaces: Holographic Entanglement Entropy Beyond the Classical Regime,

N. Engelhardt and A. C. Wall, “Quantum Extremal Surfaces: Holographic Entanglement Entropy Beyond the Classical Regime,” JHEP 01 (2015) 073, arXiv:1408.3203 [hep-th]

arXiv 2015

[8] [8]

state-dependent

demonstrated that for large enough code subspaces this intuition is false. Consider trying to reconstruct a black hole operator using a boundary region A that is barely more than half the CFT. If this black hole were in a pure state, the entanglement wedge of A would include the black hole. However, if our code subspace included, say, all eSBH black hole ...

work page