REVIEW 3 major objections 5 minor 43 references
Modifications of the Page Curve from correlations within Hawking radiation
T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash
Pith's one-line read This paper argues that correlations between successive Hawking pairs cannot restore unitarity in a Bell-pair model, because the entropy bound forces the Page curve to turn at one-fifth the black hole lifetime.
desk verdict A defensible small-correction no-go bound with some genuinely new quantitative content, but the paper's headline Page-time inference outruns what eq. (28) proves. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing object is a two-dimensional Hilbert space $V_2$ for each Hawking pair, spanned by $S^{(1)}=(|00\rangle+|11\rangle)/\sqrt{2}$ and $S^{(2)}=(|00\rangle-|11\rangle)/\sqrt{2}$; every emitted pair is a superposition of these two Bell-type states. A new emission changes the black-hole state from $|\psi_i\rangle$ to $S^{(1)}|\psi_i^{(1)}\rangle+S^{(2)}|\psi_i^{(2)}\rangle$, so the pair can be correlated with everything that came before. The proof runs through two lemmas: Lemma 1 bounds the entropy of the newly created pair, and Lemma 2 bounds the entropy of a single outgoing member. Strong subadditivity supplies the lower bound on $\Delta S$ and ordinary subadditivity supplies the upper bound, so the result depends only on the overlap parameters and survives even when the correction is of order one.
What would settle it
Extend the pair Hilbert space to $V_4$ by allowing the new pair to be a superposition that includes $|01\rangle$ or $|10\rangle$ (or give the off-diagonal overlap an imaginary part), recompute $\Delta S$, and check whether inequality (28) is ever violated; a single violation, or a unitary model in this larger space whose Page curve turns over near $0.53\,t_{\rm decay}$, would refute the paper's conclusion.
Extended reading notes
Core claim
The central result is Theorem 1 (eq. 28): each evaporation step changes the outgoing radiation's entanglement entropy by an amount $\Delta S$ obeying $\left(1-4\epsilon_2^2-\sqrt{1-\gamma^2}\right)\log 2 \le \Delta S \le \sqrt{1-4\epsilon_2^2}\,\log 2$, with $\epsilon_2$ the off-diagonal overlap between the two allowed pair states and $\gamma$ measuring the admixture of the correction. The inequality holds for corrections of any size, not just small ones. Restoring unitarity would require the lower bound to go negative, which forces $\frac{1}{2}<\epsilon<\frac{\sqrt{3}}{2}$, a large deformation. Even inside that regime, the envelope of allowed Page-like curves turns over at $t_{\rm Page}\approx 0.2\,t_{\rm decay}$, corresponding to $\beta=6.24$; Page's unitary expectation is $t_{\rm Page}\approx 0.53\,t_{\rm decay}$ with $\beta\approx 1.48472$. The authors take this as evidence that Bell-pair-only evaporation cannot be compatible with unitary evolution.
Load-bearing premise
The bounds assume each Hawking pair lives in the two-state space spanned by $|00\rangle$ and $|11\rangle$, with the off-diagonal overlap treated as real, so the proof does not cover pairs containing $|01\rangle$ or $|10\rangle$ components or complex overlaps.
Editorial extensions
If this is right
- Small corrections ($|\epsilon|\ll 1$) leave the entropy increase per step within a hair of $\log 2$, so they cannot turn the Page curve around.
- Even order-one corrections cannot make $\Delta S$ drop below the lower bound, so no emission step can decrease the entropy by more than a bounded amount.
- A unitary Page curve requires the lower bound to become negative, which forces $\frac{1}{2}<\epsilon<\frac{\sqrt{3}}{2}$, a necessarily large correction.
- The envelope of all allowed curves turns over at $t_{\rm Page}\approx 0.2\,t_{\rm decay}$, corresponding to $\beta=6.24$, far from Page's unitary expectation, so Bell-pair-only evaporation is incompatible with unitary evolution.
Reading between the lines
- Because the bound comes from subadditivity at the level of individual emissions, any model with the same two-state pair structure will inherit the same entropy band, regardless of the detailed evaporation dynamics.
- A natural next test is to add the two missing states $|01\rangle$ and $|10\rangle$: those components carry bit-flip information that a $|00\rangle/|11\rangle$ superposition cannot, and they may allow the entropy increase to be cancelled without leaving the horizon.
- The early turnover at $0.2\,t_{\rm decay}$ is an envelope statement, not a prediction of the actual entropy curve; a full model could sit anywhere inside the allowed region, so the result is best read as a no-go band for Bell-pair-only evaporation.
- One can test the bound numerically by simulating unitary evaporation on a small qubit register with fixed correction strength, computing $\Delta S$ per step, and comparing the sampled envelope with inequality (28).
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies a qubit toy model of black-hole evaporation in which each emitted Hawking pair is a superposition of two Bell-type states, S(1) and S(2). It first reproduces Mathur's conclusion that small corrections cannot stop the growth of entanglement entropy, then relaxes the smallness condition and derives a two-sided bound on the one-step entropy change, ΔS, using strong subadditivity and subadditivity (Theorem 1, Eq. (28)). The authors then use these bounds to construct an envelope for possible Page curves, read off an intersection at t_Page ≈ 0.2 t_decay corresponding to β ≈ 6.24, and conclude that Bell-pair-only evaporation is incompatible with unitary evolution.
Significance. If the central bound and the Page-curve inference were both valid, the paper would provide a quantitative extension of Mathur's small-correction analysis to O(1) corrections and would sharpen the statement that Bell-pair-type corrections cannot unitarize black-hole evaporation. The derivation of Theorem 1 is self-contained, uses standard entropy inequalities, and contains no fitted parameters; these are genuine strengths. However, the advertised Page-curve conclusion does not follow from the bound as shown, and the complexification assertion in Section V needs proof. With revision, the bound itself could stand as a modest but useful contribution; in its present form the main quantitative claim about the early Page time is not supported.
major comments (3)
- [VI, Eq. (28)] The inference from the envelope of the bounds to an actual Page time is not valid. Equation (28) bounds the one-step increment ΔS with parameters that are free to vary from step to step, and Eq. (33) is a statement about the maximum width of the allowed band, not about any particular history. A history that takes ε2=0 for the first N/2 emissions, so that ΔS=log2 at each of those steps, and then chooses parameters near the lower bound for the remaining N/2 emissions respects (28) at every step but turns over at n=N/2, i.e. t_Page≈0.5 t_decay. Section VI supplies no argument that a physical trajectory must saturate the upper bound before the intersection and the lower bound after it; consequently the claimed t_Page=0.2 t_decay and β=6.24 are not consequences of Theorem 1.
- [V, after Eq. (20)] The assertion that complexifying ε2 'gives the same result' is unproved and is not immediate. Lemma 1's γ depends on |⟨Λ(1)|Λ(2)⟩|, while the diagonal entries of ρ_{b_{n+1}} in Eq. (26) depend on Re⟨Λ(1)|Λ(2)⟩. For a generic complex overlap the two lemmas cannot be combined through a single real parameter. If a phase redefinition of S(2) renders the overlap real, that argument should be stated explicitly; as written, the theorem's claimed generality over complex overlaps is unsupported.
- [III and VI, Eq. (5)] The theorem and the Page-curve discussion are established only for the truncated pair Hilbert space V2=span{S(1),S(2)}. The paper asserts that sequences from (V2)^N possess enough complexity to encode the black-hole information, but it does not prove that the bounds in (28) or the envelope argument survive in V4 or in a general pair Hilbert space. Since the conclusions are phrased as statements about Bell-pair corrections generally, the restriction to V2 should either be lifted or else incorporated explicitly into the statement of the claims.
minor comments (5)
- [V, Eq. (20)] The notation in Eq. (20) is confusing: the same printed symbol appears to be used for the diagonal weight ⟨Λ(2)|Λ(2)⟩, the off-diagonal overlap, and the parameter ε2 in Theorem 1. The authors should introduce distinct symbols, such as ε, ε², and δ, and define their ranges clearly.
- [VI, final paragraph before Conclusion] The sentence 'This corresponds to β = 6.24' appears twice in quick succession; the duplicated phrase should be removed.
- [I, Introduction] The word 'asymtopia' should be 'asymptopia', and the manuscript would benefit from a careful proofreading pass for other typographical errors.
- [VI, Figure 1] The text references Figure 1a, 1b, and 1c, but no figure content appears in the version I reviewed; please ensure the figures are included and that the grey envelope region described in the text is visible in Figure 1c.
- [V.B, Eq. (34)] The step from the negativity condition on the lower bound in (28) to the displayed inequality (34) should be expanded; in particular, the allowed range of ε2 that justifies taking the maximum of the left-hand side should be stated.
Circularity Check
No circular reduction: the ΔS bounds are derived from explicit density-matrix diagonalization and subadditivity, not from the conclusion.
full rationale
The central derivation is self-contained. Theorem 1's bounds (eq. 28) follow from Lemmas 1 and 2, which are proven by explicit diagonalization of the reduced density matrices (21) and (26); the lemmas are then combined through strong subadditivity and subadditivity for A={b}, B=b_{n+1}, C=c_{n+1}. The correction parameters in eq. (20) are defined from the state and are not fitted to any data, and no fitted quantity is later relabeled as a prediction. Section IV's 0≤ΔS≤log2 is a direct consequence of the explicitly stated V2={|00>,|11>} truncation, an exposed model assumption rather than a hidden input. The only self-citation, [26], is a pointer to a preliminary conference version and is not load-bearing: the present paper re-proves the needed statements, so no premise reduces to the authors' prior unverified claim. §VI's t_Page=0.2 t_decay is obtained by taking the intersection of the envelope bounds, a conventional definition rather than a trajectory prediction; the paper does not show that actual evaporation histories saturate the envelope, so this is an unsupported extrapolation, not a circular reduction of Theorem 1 to its inputs. No self-definitional, fitted-input, or uniqueness-importing pattern is present.
Assumptions & free parameters
assumptions (5)
- domain assumption Hawking pairs can be modeled as qubit pairs in the two-dimensional space V2 = span{S(1),S(2)}.
- domain assumption Previously emitted b quanta are causally disconnected from later emissions, so new b emissions do not affect their entropy.
- standard math Strong subadditivity and subadditivity of von Neumann entropy.
- ad hoc to paper The off-diagonal overlap <Lambda(1)|Lambda(2)> can be taken real, and complexifying gives the same result.
- ad hoc to paper The entropy bounds can be converted into a Page-curve envelope by fixing t_decay and taking the intersection of the rising and falling envelope lines as the Page time.
Cite this review
Pith. "Pith review of Modifications of the Page Curve from correlations within Hawking radiation." pith.science (2026). https://pith.science/paper/VGF5KLXG
@misc{pith2026190809669,
author = {Pith},
title = {Pith review of: Modifications of the Page Curve from correlations within Hawking radiation},
year = {2026},
howpublished = {\url{https://pith.science/paper/VGF5KLXG}},
note = {Machine review of arXiv:1908.09669}
}
read the original abstract
We investigate quantum correlations between successive steps of black hole evaporation and investigate whether they might resolve the black hole information paradox. 'Small' corrections in various models were shown to be unable to restore unitarity. We study a toy qubit model of evaporation that allows small quantum correlations between successive steps and reaffirm previous results. Then, we relax the 'smallness' condition and find a nontrivial upper and lower bound on the entanglement entropy change during the evaporation process. This gives a quantitative measure of the size of the correction needed to restore unitarity. We find that these entanglement entropy bounds lead to a significant deviation from the expected Page curve.
Figures
Reference graph
Works this paper leans on
-
[1]
The black hole geometry can be foliated by spacelike slices continuous at the horizon and the physics on these slices is given by a local quantum field theory [10, 20]
-
[2]
Suppose a collapsing shell with state |Ψ⟩M produces the black hole. Time evolution of the hole, will push this matter far along spacelike slices inside the hole such that it is far from where Hawking radiation is being produced. Thus |Ψ⟩M will at most, weakly affect the Hawking pairs. Thus, after N Hawking pairs are radiated the black hole + radiation stat...
-
[3]
The states will be related by [21, 22], |Ψ⟩pair =Ceβc†b† |0⟩c|0⟩b
The stretching of spacelike slices will cause creation/annihalation operators on dif- ferent slices to be linearly related. The states will be related by [21, 22], |Ψ⟩pair =Ceβc†b† |0⟩c|0⟩b. (2) 4 whereβ is ac-number andc† andb† are respectively, creation operators of the ingo- ing Hawking particle that is captured by the black hole, and the outgoing Hawk...
-
[4]
(22) 9 Hence, the entanglement entropy of the pair is S(p) =−trˆρp log ˆρp =− 2∑ i=1 λi logλi = log 2− 1 2[(1 +γ) log(1 +γ) + (1−γ) log(1−γ)]. (23) It can be shown that for 0 ≤x≤ 1, (1−x2) log 2≤ log 2− 1 2[(1 +x) log(1 +x) + (1−x) log(1−x)]≤ √ 1−x2 log 2. (24) The result follows from (24). Lemma 2. (1− 4ϵ2
-
[5]
S. W. Hawking, Phys. Rev. Lett. 26, 1344 (1971)
1971
-
[6]
log 2≤S(bn+1) =S(cn+1)≤ √ 1− 4ϵ2 2 log 2 Proof. The complete state of the system after the creation of n + 1 pairs is |ΨM,c,ψb(tn+1)⟩ = [ |0⟩cn+1|0⟩bn+1 1√ 2(Λ(1) + Λ(2)) ] + [ |1⟩cn+1|1⟩bn+1 1√ 2( Λ(1)− Λ(2)) ] , (25) where Λ(1) and Λ(2) reflect the state of the black hole and are defined by (9). Now, the reduced density matrix of the cn+1 or bn+1 quanta i...
-
[7]
(29) Inequality (29) follows from using Lemma 1 and Lemma 2
log 2− √ 1−γ2 log 2. (29) Inequality (29) follows from using Lemma 1 and Lemma 2. Now using the subadditivity inequality, S(A) +S(B)≥S(A,B ), we find, S({b}) +S(bn+1)≥S({b},bn+1) ⇒ ∆S≤ √ 1− 4ϵ2 2 log 2 (30) Inequality (30) follows from Lemma 2 and combining (29) and (30) gives Theorem 1. The change in the entanglement entropy upper and lower bounds in Theo...
-
[8]
Then (31) reduces to D∆S log 2 = 2 √ 1− 4ϵ2 2− (1− 4ϵ2
Show all 43 references
-
[9]
This can also be seen from (22)
(32) It is straightforward to show that maximization of D∆S requires dD∆S dϵ2 = 0 ⇒ ϵ2 = 0. This can also be seen from (22). This gives, D∆S log 2≤ 1. (33) Thus D∆S never exceeds log 2. While it is clear that that ∆ S≤ log 2, the result that D∆S≤ log 2 is a different nontrivial...
-
[10]
The purpose is to familiarize the reader with the framework we will use
and that was followed by others. The purpose is to familiarize the reader with the framework we will use. We make certain assumptions that will allow us to use local effective field theory to handle the quantum gravity effects that produce Hawking radiation. This means that given...
-
[11]
(35) 12 This gives a necessary (but not sufficient) condition for unitarity
This implies that 1− 4ϵ2(1−ϵ2)< 1 4 =⇒ 1 2 <ϵ< √ 3 2 . (35) 12 This gives a necessary (but not sufficient) condition for unitarity. Equation (35) implies that relatively large corrections are needed. The bound in (28) does not require the cor- rection parameters to be small and ...
-
[12]
Therefore, we need to consider the minimum value of the quantity 1−4ϵ2 2− √ 1− 4ϵ2 2, which is− 1
-
[13]
Note, in drawing the graph in Figure 1c, we first fixed the evaporation time tdecay
The Page-like curve that our evaporation model generates is bounded by the grey region in Figure 1c. Note, in drawing the graph in Figure 1c, we first fixed the evaporation time tdecay. Then we imposed the two bounds. We will take the intersection of both bounds in Figure 1c to ...
-
[14]
J. D. Bekenstein, Lettere Al Nuovo Cimento (1971–1985) 4, 737 (1972)
1972
-
[15]
J. D. Bekenstein, Phys. Rev. D 7, 2333 (1973)
1973
-
[16]
Penrose and R
R. Penrose and R. Floyd, Nat. Phys. Sci. 229, 177 (1971)
1971
-
[17]
Christodoulou, Phys
D. Christodoulou, Phys. Rev. Lett. 25, 1596 (1970)
1970
-
[18]
J. M. Bardeen, B. Carter, and S. W. Hawking, Commun. Math. Phys. 31, 161 (1973)
1973
-
[19]
S. W. Hawking, Commun. Math. Phys. 43, 199 (1975)
1975
-
[20]
M. K. Parikh and F. Wilczek, Phys. Rev. Lett. 85, 5042 (2000), arXiv:hep-th/9907001
2000 arXiv
-
[21]
Preskill, in Proceedings of the International Symposium on Black Holes, Membranes, Wormholes and Superstrings, S
J. Preskill, in Proceedings of the International Symposium on Black Holes, Membranes, Wormholes and Superstrings, S. Kalara and DV Nanopoulos, eds. (World Scientific, 1992) pp. 22–39, arXiv:hep-th/9209058
1992 arXiv
-
[22]
S. D. Mathur, Classical Quant. Grav. 26, 224001 (2009), arXiv:0909.1038 [hep-th]
2009 arXiv
-
[23]
S. D. Mathur, Classical Quant. Grav. 28, 125010 (2011), arXiv:1012.2101 [hep-th]
2011 arXiv
-
[24]
S. D. Mathur and C. J. Plumberg, JHEP 2011 (2011), arXiv:1101.4899 [hep-th]
2011 arXiv
-
[25]
S. B. Giddings, Phys. Rev. D 85, 44038 (2012), arXiv:1108.2015 [hep-th]
2012 arXiv
-
[26]
S. G. Avery, JHEP 2013, 176 (2013), arXiv:1109.2911 [hep-th]
2013 arXiv
-
[27]
Polchinski, in New Frontiers in Fields and Strings (World Scientific, 2016) arXiv:1609.04036 [hep-th]
J. Polchinski, in New Frontiers in Fields and Strings (World Scientific, 2016) arXiv:1609.04036 [hep-th]
2016 arXiv
- [28]
-
[29]
D. N. Page, Phys. Rev. Lett. 71, 1291 (1993), arXiv:gr-qc/9305007. 16
1993 arXiv
-
[30]
D. N. Page, Phys. Rev. Lett. 71, 3743 (1993), arXiv:hep-th/9306083
1993 arXiv
-
[31]
D. N. Page, J. Cosmol. Astropart. P. 2013, 28 (2013), arXiv:1301.4995 [hep-th]
2013 arXiv
-
[32]
S. D. Mathur, Lecture Notes in Physics 769, 3 (2009), arXiv:0803.2030 [hep-th]
2009 arXiv
-
[33]
S. W. Hawking, Phys. Rev. D 14, 2460 (1976)
1976
-
[34]
S. B. Giddings and W. M. Nelson, Phys. Rev. D 46, 2486 (1992)
1992
-
[35]
M. K. Parikh, in Proceedings of the MG10 Meeting held at Brazilian Center for Research in Physics (CBPF) (2006) pp. 1585–1590, arXiv:hep-th/0402166
2006 arXiv
-
[36]
Arzano, A
M. Arzano, A. J. M. Medved, and E. C. Vagenas, JHEP 2005, 37 (2005), arXiv:hep- th/0505266
2005
-
[37]
Zhang, Q
B. Zhang, Q. yu Cai, M. sheng Zhan, and L. You, Ann. Phys. 326, 350 (2011), arXiv:0906.5033 [hep-th]
2011 arXiv
-
[38]
A. Roy, M. H. Rahat, M. A. Alvi, M. Majumdar, and A. Matin, in 2012 7th International Conference on Electrical and Computer Engineering (2012) pp. 767–770
2012
-
[39]
S. K. Foong and S. Kanno, Phys. Rev. Lett. 72, 1148 (1994)
1994
- [40]
-
[41]
Sanchez-Ruiz, Phys
J. Sanchez-Ruiz, Phys. Rev. E 52, 5653 (1995)
1995
-
[42]
Bradler and C
K. Bradler and C. Adami, Phys. Rev. Lett. 116, 101301 (2016), arXiv:1505.02840 [quant- ph]
2016 arXiv
-
[43]
S. B. Giddings and Y. Shi, Phys. Rev. D87, 064031 (2013), arXiv:1205.4732 [hep-th]. 17
2013 arXiv
Reviewed August 14, 2026 · model on record in the stance chip above.
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