Comparative Requirements for Information-Recovery Frameworks

A durable comparison of information-recovery frameworks must preserve the type of every claim. An exact theorem, a controlled model, a consistency condition, a conjecture, an unknown, and an operational prediction are different kinds of knowledge. This page combines the chapter’s requirements into one matrix and asks what derivation, adverse check, error bound, or observable would satisfy each cell. It does not rank proposals with a single score: a framework can be strong on microscopic unitarity, conditional on its interior reconstruction, and silent on feasible verification.

Required background. Quantum Information for Black-Hole Physics supplies channels, entropy, recovery, and error metrics.

Helpful background. Black-Hole Thermodynamics and Semiclassical Gravity supplies generalized entropy and backreaction; Quantum Fields in Curved Spacetime supplies the horizon EFT baseline; Black-Hole Information supplies the results being compared; Quantum Gravity supplies candidate microscopic completions.

The comparison record

For every framework, record the same objects:

$$\mathcal C= (\text{claim},\text{system},\text{state},\text{regime}, \text{derivation},\text{error},\text{adverse check},\text{test}).$$

The claim is one sentence that could be false. The system includes Hilbert space or observable algebra and all endpoint sectors. The state and regime fix semiclassical, large-entropy, code-subspace, replica, or protected-sector assumptions. The derivation has an exact source locator. The error names a norm and scaling. The adverse check changes a consequential assumption. The test gives a probability or bounded observable that differs from a competitor.

Missing entries are marked unknown, not inferred from the framework’s motivation. A claim that is meaningful only after ensemble averaging is marked ensemble result, not silently upgraded to one-theory unitarity. A recovery map that exists but has no efficient algorithm is marked existence, not operationally feasible.

Six-axis comparison of representative frameworks

The table summarizes the strongest conservative statement supported by the named approach. Each cell remains conditional on the detailed page and source; it is not a substitute for those derivations.

Framework or proposal	Unitarity and entropy	Horizon and EFT	Interior and causality	Factorization and microscopic definition	Dynamics and errors	Operations and tests
Semiclassical Hawking evolution with complete evaporation, Hawking 1976, §§II–IV, pp. 2462–2471	Leading exterior calculation gives thermal flux and rising radiation entropy; a pure-to-mixed endpoint follows if no additional output is admitted	Controlled for low-curvature exterior observables in the stated state and adiabatic regime	Conventional smooth horizon initially, but no unitary late-time interior completion	Effective field theory, not a microscopic one-theory completion	Predicts flux and stress tensor, but no correlation-restoring late-time channel under its complete-endpoint assumptions	Exterior flux is observable; a fine-grained recovery test would require many preparations or microscopic access not supplied by the approximation
Replica wormholes and islands, Almheiri et al. 2020, §§2–6	Controlled Page-like generalized entropy in selected bath and holographic models	Can retain local smooth-horizon EFT while changing the dominant entropy saddle	Island may enter the radiation wedge; recovery exists on a declared code subspace	JT amplitudes admit specified matrix-integral ensemble completions, while the bath-island entropy remains a semiclassical result; the baby-universe interpretation is analyzed in Marolf and Maxfield 2020, §§2–5	Entropy transitions are calculable, while individual amplitudes and Lorentzian information transfer are not fixed by the entropy formula alone	Decoder cost and direct verification remain unknown in the generic regime
Holographic error correction, Almheiri, Dong, and Harlow 2015, §§2–4 and Dong, Harlow, and Wall 2016, complete Letter and proof	Boundary evolution is unitary in a fixed CFT and bulk entropy is reproduced in controlled large-N settings	Smooth bulk EFT within a semiclassical code subspace	Multiple boundary representations can encode one logical interior operator without cloning	Fixed boundary theory is defined, but bulk reconstruction is subspace- and order-dependent	Exact recovery follows under the finite theorem’s hypotheses; approximate gravitational recovery needs a separately derived norm conversion	Efficient decoding, coherent radiation access, and a feasible bulk verification protocol remain separate requirements
String microstate and fuzzball programs, Bena et al. 2022, §§2–8	Exact state counts and unitary microscopic dynamics exist in important protected sectors	Some constructions replace or modify the conventional horizon; behavior of typical states is disputed	Interior may be emergent, state-specific, or absent depending on the proposal	Fixed string theory supplies a microscopic setting for the constructed sectors	Generic collapse, radiation dynamics, and approximation error remain incomplete	Distinctive observations require predictions stable under source populations, propagation, and detector uncertainty
Soft nonlocal information transfer, Giddings 2013, §§2–5	Introduces effective atmosphere-scale couplings designed to transfer correlations with modest energy flux	Modifies locality over an atmosphere-scale region rather than imposing a Planckian wall	Interior consistency depends on the nonlocal algebra and causal signaling bounds	Usually an effective interaction class rather than a UV definition	Transfer rates and exterior response can be estimated once the interaction and state are fixed	A useful test must distinguish the transfer from ordinary environmental or near-horizon effects within its time, energy, and backreaction budget
Final-state or postselected interior, Horowitz and Maldacena 2004, §§2–3 and Gottesman and Preskill 2004, complete critique	Can induce a teleportation-like conditional map with specially entangled final data	May leave much of exterior EFT unchanged before the endpoint	Replaces ordinary interior evolution by a final boundary condition	Requires a microscopic rule for postselection and normalization	Generic interior interactions spoil the ideal map unless additional tuning is imposed; linearity, signaling, and normalization are decisive adverse checks	The proposal must specify probabilities available to exterior observers without assuming access to the postselected interior outcome

The matrix shows why no row receives a blanket label of “solution.” The island row contains a strong controlled entropy result; the holographic row contains a strong fixed-theory unitarity result; the microstate row contains strong constructions in selected sectors. Their unresolved cells are not the same. The modern entropy review explicitly distinguishes a Page curve from individual matrix elements in Almheiri et al. 2021, §§8–11, while the quantum-information review separates recoverability from decoding complexity in Harlow 2016, §§5.4–5.7 and Appendix E.

What would satisfy each requirement?

Requirement	Sufficient evidence in the claimed regime	Strong adverse check
Unitarity and fine-grained entropy	Complete unitary or S-matrix preserving reference entanglement, charges, and final purity, together with a density operator or controlled replica derivation for the declared radiation algebra	Include soft, remnant, baby-universe, and endpoint sectors, test superpositions, and construct inequivalent channels with the same entropy curve
Horizon EFT	Dressed detector and stress-tensor bounds over stated distances, frequencies, times, and states	Accumulate the proposed corrections over an old hole and test backreaction
Interior consistency	Operator domain, commutators, reconstruction norm, causal access, and endpoint that pass no-cloning and overlap tests	Enlarge the code, superpose reference states, or close a duplication protocol
Factorization	Nonperturbative fixed-theory definition reproducing connected saddles while preserving independent-boundary products	Compare an ensemble covariance with a genuine one-member observable
Dynamics and errors	Real-time channel or amplitudes with convergence, a named error norm, and a controlled first omitted term	Change timestep, regulator, saddle family, state, or coupling and test whether the claimed transfer survives
Operations and tests	Decoder, copies, collection efficiency, memory, gates, time, energy, backreaction, and a distinguishable outcome distribution	Require the protocol to meet its actual causal and storage deadlines; an interior comparison must precede its endpoint, while a radiation-only test may occur later

A theorem can satisfy a row without an experiment if the claim is mathematical. An operational claim needs more: a realizable protocol and an outcome model. Conversely, a null astronomical observation cannot decide microscopic unitarity unless competing frameworks predict different likelihoods after source and instrument effects. Generic detector inference and explicit string construction are treated in their own volumes; this page preserves only the requirements they must export to the information question.

Avoid scalar scores and hidden assumptions

Assigning points to the matrix would create false precision. Unknown evidence is not a failed consistency test, and a theorem in a narrow regime is not equal to a qualitative universal claim. Instead, compare frameworks cell by cell using partial order: framework $F_1$ is better supported than $F_2$ on one axis only when both address the same claim and $F_1$ has stronger evidence under no stronger assumptions.

Assumptions should be visible as columns, not footnotes. Examples include a nongravitating bath, replica symmetry, analytic continuation, a fixed alpha sector, large $N$, a small code subspace, Haar randomness, protected charge, postselection, or atmosphere-scale nonlocality. If changing one reverses the conclusion, the matrix records that dependence rather than averaging it away.

Model-specific evidence must also retain its direction. JT gravity can show how wormholes and an ensemble fit exactly; it cannot establish that every CFT is averaged. Smooth microstate geometries show that rich horizon-scale states exist; they do not establish that known solutions are typical. Explicit nonlocal couplings show consistency targets; they do not supply a microscopic derivation. These boundaries let genuine progress remain visible without premature universality.

Results and open questions through 2026-08-19

Cross-cutting claim	Assumptions	Controlled result	Limitation or unresolved comparison
Fine-grained Page behavior can arise gravitationally, Almheiri et al. 2020, §§2–6	Specified low-dimensional bath model, state, QES candidates, replica saddles, contour, and continuation	Reproduces a Page-like generalized-entropy turnover in the stated approximation	One-theory amplitudes, contour uniqueness, and broad four-dimensional control remain separate
Interior operators can be redundantly encoded, Dong, Harlow, and Wall 2016, complete Letter and proof	Finite regulated code, declared logical algebra and boundary region, and relative-entropy hypotheses	Establishes multiple physical representatives of one logical operator on the code	Large-domain, nonperturbative, state-overlap, and efficient-decoding questions remain open
Connected topologies reproduce ensemble correlations, Saad, Shenker, and Stanford 2019, §§2–5	Perturbative JT topology expansion and a specified double-scaled matrix-integral completion	Matches connected topology terms to connected ensemble observables order by order	Does not make a universal claim about factorization in every fixed boundary theory
Reflection-positive QFT reconstruction, McNamara and Wang 2026, §1.1, Theorem 1.1; §§5.2–5.3, 9, and 11	Finite, multiplicative, reflection-positive closed-manifold partition function in the compact source class, with the full charged spectrum	Reconstructs a unitary compact functorial QFT and attributes apparent Hilbert-space nonfactorization to omitted charged states	Application to gravity requires proof of every hypothesis and does not construct the gravitational UV completion
Microscopic state counting and geometries, Bena et al. 2022, §§2–8	Protected string-theory sectors and explicitly constructed state families or geometries	Supplies protected entropy counts and many nontrivial microstate constructions	Typical-state geometry and generic real-time evaporation remain proposal-dependent
Operational verification is more demanding than recoverability, Harlow 2016, §§5.4–5.7 and Appendix E and Acharya et al. 2020, §§1.3, 4, and 6–7	Declared circuit and access model, finite-dimensional state, independent-copy supply, accuracy, and success probability	Complexity and copy bounds quantify two distinct resource obstructions	No single theorem closes copies, gates, time, energy, collection, dressing, and backreaction for every protocol

The strongest competing interpretations differ mainly on how far controlled model results can be exported: some view the combined holographic, island, and error-correction picture as a compelling mechanism; others require a fixed- theory Lorentzian observable and nonperturbative errors before making that claim. The full-spectrum reconstruction theorem supplies a third position under formal-QFT hypotheses: apparent nonfactorization can reflect omitted charged states rather than an ensemble. The conclusions could change if such a calculation appears, if an explicit counterexample defeats a central reconstruction or factorization assumption, or if a feasible experiment distinguishes two rows while respecting its own resource and backreaction budget.

Check your understanding

A framework has a fixed unitary boundary Hamiltonian and establishes a JLMS-type relative-entropy mismatch $\lvert\Delta_{\mathrm{JLMS}}\rvert=O(1/N)$ on a declared code. Which matrix cells are supported, and which remain conditional? The relative-entropy statement is derived in Jafferis et al. 2016, §§2–4.

Solution

Fixed-theory unitarity is supported for the boundary state family, and perturbative relative-entropy matching is supported for the stated code, region, regulator, and central data. The raw $O(1/N)$ mismatch does not automatically become an $O(1/N)$ operator-norm reconstruction error. That stronger cell needs a quantitative recovery theorem with fixed complementary data and an explicit conversion to the chosen channel or operator norm. One must still identify the radiation algebra and derive its fine-grained entropy, show the horizon EFT regime, control nonperturbative errors as the code grows, relate the setup to evaporation, and supply a decoder with time, memory, collection, and backreaction costs. Those cells remain conditional or unknown; they are not failures of the two established results.

References

• Acharya, Jayadev, Ibrahim Issa, Nirmal V. Shende, and Aaron B. Wagner, “Estimating Quantum Entropy,” IEEE Journal on Selected Areas in Information Theory 1 (2020) 454–468. DOI. Open manuscript
• Almheiri, Ahmed, Thomas Hartman, Juan Maldacena, Edgar Shaghoulian, and Amirhossein Tajdini, “Replica Wormholes and the Entropy of Hawking Radiation,” Journal of High Energy Physics 05 (2020) 013. DOI. Open manuscript
• Almheiri, Ahmed, Thomas Hartman, Juan Maldacena, Edgar Shaghoulian, and Amirhossein Tajdini, “The Entropy of Hawking Radiation,” Reviews of Modern Physics 93 (2021) 035002. DOI. Open manuscript
• Almheiri, Ahmed, Xi Dong, and Daniel Harlow, “Bulk Locality and Quantum Error Correction in AdS/CFT,” Journal of High Energy Physics 04 (2015) 163. DOI. Open manuscript
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• Giddings, Steven B., “Nonviolent Nonlocality,” Physical Review D 88 (2013) 064023. DOI. Open manuscript
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• Horowitz, Gary T. and Juan Maldacena, “The Black Hole Final State,” Journal of High Energy Physics 02 (2004) 008. DOI. Open manuscript
• Jafferis, Daniel L., Aitor Lewkowycz, Juan Maldacena, and S. Josephine Suh, “Relative Entropy Equals Bulk Relative Entropy,” Journal of High Energy Physics 06 (2016) 004. DOI. Open manuscript
• Marolf, Donald and Henry Maxfield, “Transcending the Ensemble: Baby Universes, Spacetime Wormholes, and the Order and Disorder of Black Hole Information,” Journal of High Energy Physics 08 (2020) 044. DOI. Open manuscript
• McNamara, Jacob and Zhencheng Wang, “Wormholes as Red Herrings: Reflection Positivity and the Reconstruction of Unitary Quantum Field Theories,” arXiv:2607.01322 (2026). Open manuscript
• Saad, Phil, Stephen H. Shenker, and Douglas Stanford, “JT Gravity as a Matrix Integral,” arXiv:1903.11115 (2019). Open manuscript