Islands for black holes in a hybrid quantum state

Debajyoti Sarkar; Sergey N. Solodukhin; Yohan Potaux

REVIEW 3 major objections 3 minor 1 cited by

Islands for black holes in a hybrid quantum state

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Pith's one-line read For a hybrid Unruh/Boulware state with negative total central charge, radiation entanglement entropy follows a Page curve even before islands are included, and the island entropy is always larger, so minimality selects the no-island curve.

desk verdict A technically careful island calculation whose central claim rests on using the standard CFT entropy formula for negative central charge, producing negative entropies that diverge to minus infinity. read the letter →

arxiv 2411.09574 v1 pith:22FGRWQB submitted 2024-11-14 hep-th gr-qc

Yohan Potaux , Debajyoti Sarkar , Sergey N. Solodukhin This is my paper · ORCID

classification hep-thgr-qc MSC 81T2083C5781T4083C80 PACS 04.70.Dy04.62.+v

keywords blackholeinformationparadoxPagecurveentanglemententropyislandstwo-dimensionaldilatongravityhybridquantumstatesnegativecentralchargeghostfields

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

Black hole information loss is usually diagnosed by whether radiation entanglement entropy first rises and then falls, a Page curve. This paper studies a two-dimensional semi-classical gravity model in a hybrid quantum state: physical fields in the Unruh state plus non-physical ghost fields in the Boulware state, with negative total central charge $\kappa<0$. It computes the entanglement entropy of the radiation and finds that along any fixed incoming null line the change in entropy $\Delta S_{\rm ent}=4\kappa\rho$ rises to a maximum and then falls, so the fine-grained entropy already follows a Page curve. It then applies the island prescription and finds that the generalized entropy of the island is always larger than the entropy without an island, so the minimum selects the no-island result. If correct, this independently confirms an earlier thermodynamic-entropy computation and suggests that islands are not needed when ghost fields with negative central charge are present.

What carries the argument

The load-bearing object is the conformal-factor entanglement entropy formula $S_{\rm ent}=4\kappa(\rho_O+\ln L/\epsilon)$, where $\rho$ is the conformal factor in asymptotically flat coordinates and $\kappa=N/24$; in the hybrid state $\kappa=\kappa_1+\kappa_2<0$ because the ghost fields contribute negatively to the central charge. Since $\Delta S_{\rm ent}=4\kappa\rho$, the negativity of $\kappa$ flips the sign of the entropy relative to $\rho$: at past infinity $\rho\to 0^-$, while near $x^-=0$ one has $\rho\sim(\kappa_1/2\kappa)\ln(-\lambda x^-)\to +\infty$, so a maximum appears in between. The island machinery is the generalized entropy functional (5.5), extremized over the location of the quantum extremal surface; with $\kappa<0$ the extremal surface lies on the reflected apparent-horizon curve, but its on-shell value $S_{\rm gen}=2m/\lambda-\kappa\lambda\tilde\sigma^-_O$ is always above the no-island entropy, so the minimum in the island prescription discards the island.

What would settle it

Construct the reduced density matrix of the radiation subsystem in the hybrid state and compute its von Neumann entropy directly; if the result is not $4\kappa(\rho+\ln L/\epsilon)$ or is negative where $\Delta S_{\rm ent}$ is decreasing, the claimed Page curve is not the true entanglement entropy. A simpler check is to test a lattice or exact two-dimensional model in which only the ghost sector is present and see whether its entanglement entropy is non-negative and follows the same sign-flipped curve.

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Extended reading notes

Core claim

The paper's central claim is that, for the hybrid Unruh/Boulware solution with negative total central charge, the entanglement entropy of black hole radiation follows a Page curve without any island. Along a null line $x^+=\mathrm{const}$, the change in entanglement entropy $\Delta S_{\rm ent}=4\kappa\rho$ starts at zero at past infinity, grows to a maximum on the curve (4.8), then decreases and crosses zero on the curve (4.9), tending to $-\infty$ as $x^-\to 0$. At future infinity the limit is $\Delta S^\infty_{\rm ent}=-2\kappa\ln(1-x^-_h/x^-)$, a monotonically decreasing function because $\kappa<0$. The island computation gives, at leading order, $S_{\rm gen}=2m/\lambda-\kappa\lambda\tilde\sigma^-_O$, which is always larger than the no-island branch $2\kappa\lambda\tilde\sigma^-_O$ when $\kappa<0$, so the equilibrium entropy $S_{\rm isl}=\min(\ldots)$ chooses the no-island curve. The authors conclude that the Page curve is already present in the entanglement entropy alone, and that islands do not contribute to the generalized black hole entropy here.

Load-bearing premise

The load-bearing assumption is that the formula $S_{\rm ent}=4\kappa(\rho+\ln L/\epsilon)$ remains a legitimate, non-negative entanglement entropy when the total central charge $\kappa=\kappa_1+\kappa_2$ is negative, even though the ghost fields' density matrix is not manifestly positive; the Page curve is produced entirely by that negative sign.

Editorial extensions

If this is right

In this hybrid model the fine-grained entropy of the radiation already has the rising-then-falling shape of a Page curve, so unitary information recovery does not require an island.
When the island prescription is applied, the generalized entropy of the island is always larger than the no-island entropy, so the minimum selects the no-island branch and the island has no effect.
The entanglement-entropy computation reproduces, by an independent route, the earlier thermodynamic-entropy result that the radiation entropy follows a Page curve; this is a consistency check internal to the model.
The negative outgoing energy flux of the non-physical fields is the mechanism behind the decrease: for the purely physical Boulware case with $\kappa>0$ the same method gives a monotonically increasing entropy and islands are necessary for a Page curve.
For an observer at finite future null infinity the maximum of the entropy occurs at a finite retarded time; only in the exact limit $x^+\to\infty$ does the maximum recede to past infinity.

Reading between the lines

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

If the ghost sector is interpreted as the re-emerging partner modes of the thermal radiation, this model gives a concrete two-dimensional setting in which information recovery is carried by negative-energy flux rather than by islands; a testable signature is non-zero mutual information between early and late radiation.
The paper does not reconstruct the radiation reduced density matrix; a direct computation of its von Neumann entropy would either confirm $S_{\rm ent}=4\kappa(\rho+\ln L/\epsilon)$ or show that the negative-central-charge entropy is not a physical entropy, which is the most direct check of the central claim.
In other two-dimensional or higher-dimensional semiclassical models, the same cancellation could be engineered by adding a negative-central-charge sector; the prediction would be that islands remain subdominant whenever the total central charge is negative.
The late-time branch of the selected entropy decreases without bound in the regulated variables (to $-\infty$ as $x^-\to 0$), which means the model's Page curve is only a qualitative signal of information recovery; interpreting that branch as a true entropy will require accounting for the negative values.

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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

What you should know: this paper adapts the Hartman-Shaghoulian-Strominger island computation to the hybrid Unruh/Boulware state in the RST model, with total central charge κ = κ1 + κ2 < 0. The section 3 check against the known Boulware results is solid, and the algebra in sections 4 and 5 appears consistent. But the central claim—that the fine-grained radiation entropy follows a Page curve without islands—does not hold as stated, for the reason the stress-test note identifies.

The new calculation is real: the hybrid-state entanglement entropy and its island analysis are not in the earlier papers, which treated thermodynamic entropy or the pure Boulware case. The authors also deserve credit for clear exposition, for reproducing prior results as a check, and for not fitting any parameters. The self-citation is legitimate: the master equation (4.1) comes from their earlier work, and the island computation follows [16] closely.

The soft spot is load-bearing. Equation (2.3), Sent = 4κ(ρ + ln L/ε), is the standard result for unitary CFT fields with positive central charge. Here it is applied to κ < 0, a ghost-dominated sector. The resulting ΔSent rises, peaks, crosses zero, and then diverges to −∞ (Eqs. 4.7 and 4.10). A von Neumann entropy cannot be negative or unbounded below. The decrease that produces the Page-curve shape is entirely a sign effect of negative κ, and the paper gives no positive-semidefinite density matrix or other justification for assigning a meaningful entanglement entropy to the ghost sector. The island result (5.38)–(5.39) inherits the problem: the min prescription selects the negative no-island branch, so the statement that islands are unnecessary is not supported. Also, the spacetime does not actually evaporate—the apparent horizon extends to null infinity—so calling the curve a Page curve is misleading even before the negativity issue.

This is not a small interpretational caveat; it undermines the abstract's main claim. But the paper is coherent on its own terms, and the issue is addressable in revision. It deserves a serious referee: the calculation is detailed, the literature engagement is honest, and if the authors can provide a valid ghost-sector entropy definition or reframe the claims, the result could be a useful cautionary example in the island literature. As is, I would not cite it as evidence for a no-island Page curve.

Recommendation: send to peer review, but the referee should require the authors to confront the negative-entropy problem head-on. Right now the central conclusion should not be accepted as stated.

Referee Report

3 major / 3 minor

Summary. The paper analyzes the entanglement entropy of radiation in a hybrid Unruh/Boulware state of the two-dimensional RST model, in which physical fields have positive central charge κ1 and non-physical 'ghost' fields have negative central charge κ2, with total κ = κ1 + κ2 < 0. Using the standard CFT entanglement formula Sent = 4κ(ρ + ln L/ε), the authors find that along a fixed x+ null line the entropy change ΔSent rises from zero to a maximum and then decreases, which they identify as a Page curve. They then apply the island prescription and find that, because κ < 0, the no-island branch is always selected over the island branch, so islands play no role. The main technical problem is that the decreasing branch is negative and diverges to −∞, and the island comparison inherits this issue.

Significance. The paper is careful and explicit in its algebra. It reproduces the known Boulware-state results of Hartman et al. in Section 3, including the matching of Eqs. (3.13)–(3.15) with [16], and it carries out the island extremization in detail. If the negative-central-charge entanglement entropy were a genuine entanglement entropy, the model would provide a clean example where the island prescription is unnecessary and a Page-curve-like behavior arises without islands. However, the central object is precisely the contested one: the quantity whose sign produces the decrease is the same negative-entropy branch that is later selected by the min prescription. The significance therefore depends entirely on whether Eq. (2.3) is valid for a ghost sector with κ < 0, which the manuscript neither proves nor discusses.

major comments (3)

[Sec. 4, Eqs. (4.7) and (4.10)] The claimed Page curve is not a Page curve. Along a fixed x+ line, Eq. (4.7) gives ΔSent → −∞ as x− → 0−, and at future infinity Eq. (4.10) is monotonically decreasing from 0 to −∞ for κ < 0. A von Neumann entropy is non-negative and a Page curve for unitary evaporation must return to zero. The decrease is produced solely by multiplying the standard positive-central-charge formula (2.3) by κ < 0. The manuscript does not supply a reduced density matrix for the ghost sector nor a derivation of (2.3) for negative central charge. Since this negative branch is what is later identified as the Page curve and selected by the min prescription, this is not an interpretive subtlety but the central technical assumption of the paper.
[Sec. 5.2, Eqs. (5.38) and (5.39)] The island conclusion inherits the same problem. For κ < 0, the no-island branch 2κλσ̃O is negative and decreasing, while the island branch 2m/λ − κλσ̃O is positive and increasing, so the min in Eq. (5.39) always selects the negative branch. Thus the statement that islands do not contribute is equivalent to the validity of the negative no-island entropy. If Eq. (2.3) is not applicable to the hybrid state, the min-prescription result is unsupported. If it is applicable, the quantity is not a physical entanglement entropy, and the comparison in Eq. (5.39) does not establish a unitary Page curve.
[Sec. 5.2, Eq. (5.37)] The perturbative island solution cannot be carried over from the Boulware case by simply replacing κ with a negative value. Eq. (5.37) states that e^{λσ+_Q} = λx+_Q = (3κ/2)e^{λσ̃O}(1 + O(...)); for κ < 0 the right-hand side is negative, while the left-hand side is positive by construction. Hence the claimed quantum extremal surface does not exist in the perturbative regime considered, and Eq. (5.38) for Sgen is not justified. The extremization must be rederived with the sign of κ taken into account, and all subsequent conclusions about the island branch need to be re-examined.

minor comments (3)

[Sec. 3, first paragraph] There are missing spaces in the opening sentence: 'Wewillstartbycomputingtheentanglemententropyforthesolutiondescribingphysicalfieldsin theBoulwarestate.' Similar spacing issues appear in the surrounding text.
[Sec. 5.1, Eqs. (5.15) and following paragraph] The condition for the island curve to be partially outside the singularity is written as m/λ > κ, and a few lines later the small-mass case is described as 'm < λκ'. The manuscript should use a single dimensionless form, such as m/λ > κ and m/λ < κ, to avoid confusing κ and λκ.
[Fig. 6 and Sec. 4, Eq. (4.10)] Figure 6 labels the curve a 'typical behavior for the Page Curve', but the asymptotic result in Eq. (4.10) shows that for an observer at future infinity the maximum is pushed to past infinity and the entropy is monotonically decreasing. The caption and the text should reconcile the finite-x+ behavior with the asymptotic behavior, since the latter does not exhibit the rising part of a Page curve.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Page-curve shape is solved from the adopted master equation and the standard entanglement formula, not fitted or imported as a conclusion.

full rationale

The derivation chain is not circular. The background hybrid solution and its master equation (4.1) are taken from the authors' prior work [13], but the new quantities are then computed, not assumed: the entanglement entropy is obtained by applying the standard formula Sent = 4κ(ρ + ln L/ε) (Eq. (2.3)) to the solution, and the sign of ΔSent is derived from the sign of ρ and of the total central charge κ. The apparent-maximum-then-decrease behavior along null lines follows from the asymptotic limits (4.4) and (4.6)-(4.7) and from the extremum condition (4.8); it is not obtained by fitting a Page-curve ansatz, nor is the curve defined in terms of the desired conclusion. The island analysis likewise solves the extremization conditions (5.30)-(5.31) and evaluates Sgen at the resulting saddle, giving Eq. (5.38); the min prescription (5.39) then selects the no-island branch because of the sign of κ. No parameter is fitted to a subset of data and renamed a prediction, and no uniqueness theorem or ansatz is imported from the authors' own papers to force the result. The cited prior work [1,13] provides the hybrid solution and a previous thermodynamic-entropy result, but the entanglement and island computations here are self-contained algebraic consequences of the stated equations. The physical concern that κ < 0 leads to negative and unbounded-below entanglement entropy values (e.g., Eq. (4.7)) is a question about the validity or interpretation of the input formula for ghost fields, not a circularity in the derivation.

Assumptions & free parameters 2 free parameters · 4 assumptions · 1 invented entities

The central claim rests primarily on two background formulas from the literature (entanglement entropy formula and island prescription), the master equation of the hybrid state from earlier work by the same authors, and the unstated extension of both formulas to negative central charge. No parameter is fitted to data.

free parameters (2)

Total central charge κ = κ1 + κ2 = negative (κ < 0)
The sign of the total central charge is the key input: the entire Page-curve behavior and the no-island conclusion follow from κ < 0. The value is chosen by assuming ghost fields dominate, not derived.
Shock mass m = large mass limit m/λ >> 1
The perturbative island computation and the final comparison (Eqs 5.19, 5.37-5.39) assume m/λ is large; the conclusions about islands are only established in this regime.

assumptions (4)

domain assumption Entanglement entropy of a collection of N massless scalar fields on a causal diamond in 2D conformal gauge is Sent = (N/6)(ρ_O + ln L/ε), hence 4κ(ρ+ln L/ε).
Taken from [16,18] (Eq 2.2 of the paper). Standard result for positive central charge CFTs; the paper extends it to negative central charge without re-derivation.
domain assumption The island prescription: S_isl(P_O) = min ext_{P_Q} S_gen(I ∪ R), with S_gen given by Eqs 5.3-5.5.
Assumed from the quantum extremal surface / island literature [9,11,22]. The paper applies it directly to the hybrid state.
domain assumption The hybrid state master equation (4.1) describes physical Unruh fields (central charge κ1>0) and ghost Boulware fields (κ2<0), with κ=κ1+κ2<0.
This master equation is taken from the authors' earlier paper [13] and is the starting point of both entropy and island computations. The physical content of the 'hybrid' state is not derived here.
ad hoc to paper The same entropy and quantum extremal surface formulas apply when κ < 0 (non-unitary ghost sector).
No independent justification is given that a system with negative central charge possesses a well-defined, non-negative entanglement entropy or that the island variational principle remains valid. Entered silently at Eq (2.3) and Section 5.

invented entities (1)

Non-physical (ghost) fields with negative central charge, in the Boulware state
purpose: Their presence makes the total central charge κ negative, changing the sign of the radiation entropy and making the entanglement entropy decrease after a maximum, producing a Page curve without islands.
Introduced in the authors' previous work [1,13]; no external or falsifiable evidence is provided outside the model. The negative-energy flux from these fields is the key ingredient, as stated in the conclusion.

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Pith. "Pith review of Islands for black holes in a hybrid quantum state." pith.science (2026). https://pith.science/paper/22FGRWQB

@misc{pith2026241109574,
  author       = {Pith},
  title        = {Pith review of: Islands for black holes in a hybrid quantum state},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/22FGRWQB}},
  note         = {Machine review of arXiv:2411.09574}
}

read the original abstract

Following our previous work on hybrid quantum states in the RST model, we study its most interesting solution representing a completely regular spacetime with the structure of causal diamond, containing an apparent horizon and radiation at infinity. Adapting recent computations of radiation entropy in terms of the entropy of entanglement, we find that this entropy follows a Page curve. This confirms our previous result [1], which was obtained by directly calculating the thermodynamic entropy of radiation at infinity. We also investigate the presence of a possible island in these systems, and find that it does not seem to play a role in contributing to the generalized black hole entropy.

Figures

Figures reproduced from arXiv: 2411.09574 by the authors.

**Figure 2.** An everywhere spacelike or null slice in a causal diamond. On the whole slice the [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Spacetime for the solution describing physical particles in the Boulware state. At [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

Figures from the paper (8 more)

**Figure 4.** Figure 4: Evolution of the entanglement entropy along a null line [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]

**Figure 5.** Figure 5: Spacetime for the hybrid Boulware-Unruh solution, an apparent horizon forms after [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]

**Figure 6.** Figure 6: Schematic plot of the change in entanglement entropy along an incoming null line [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]

**Figure 5.** Figure 5: figure 5. However, for an observer close to future infinity, but at a finite point, this maximum [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]

**Figure 7.** Figure 7: A full Cauchy surface that is everywhere spacelike or null, separated in three parts. [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]

**Figure 8.** Figure 8: The so-called island curve in blue indicates where the quantum extremal surface [PITH_FULL_IMAGE:figures/full_fig_p016_8.png]

**Figure 9.** Figure 9: Plot of the function f(x − Q) = x − O . We see that there is no island for x − O < 5x − h and that there are two possible islands for x − O > 5x − h . These two equations can be rewritten as λσ+ Q = λσ˜ − O + ln 3κ 2 + O e −λσ˜ − O λx− h , −λσ− Q = ln(−λx− h ) − e …

**Figure 10.** Figure 10: The island curve for the hybrid Unruh/Boulware state solution. [PITH_FULL_IMAGE:figures/full_fig_p019_10.png]

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

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