arxiv: 2511.02903 · v3 · submitted 2025-11-04 · ✦ hep-th · gr-qc

Entanglement inequalities, black holes and the architecture of typical states

Radouane Gannouji , Ayan Mukhopadhyay , Nicolas Pinochet This is my paper

Pith reviewed 2026-05-18 00:52 UTC · model grok-4.3

classification ✦ hep-th gr-qc

keywords holographic CFTAraki-Lieb inequalitytypical pure statesAdS black holesentanglement wedgeseigenstate thermalizationlength scalesfactorization

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The pith

Typical pure states in large-N holographic CFTs have two characteristic length scales fixed solely by energy and conserved charges, with effective factorization between UV and IR sectors.

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

The paper uses holographic versions of the Araki-Lieb inequality to argue that typical pure states in large N holographic conformal field theories are organized around two length scales set only by the state's total energy and conserved charges. A microscopic ultraviolet scale L_UV and a larger infrared scale L_IR divide the degrees of freedom so that those in the intermediate range factorize, one set purifying the short-distance physics and the other purifying the long-distance sector. The portion of the state that includes the ultraviolet degrees of freedom is fixed by energy and charges alone, up to corrections that are exponentially small in the entropy. This structure lets every black hole in anti-de Sitter space be separated from an asymptotic corona built from saturated entanglement wedges, with an effective buffer region appearing between the corona and the horizon.

Core claim

Using holographic realizations of the Araki-Lieb inequality, typical pure states in large N holographic CFTs possess two characteristic length scales determined solely by energy and conserved charges: a microscopic L_UV and an infrared L_IR > L_UV. Degrees of freedom between these scales effectively factorize, one purifying the ultraviolet sector and the other the infrared sector. The pure state factor including the ultraviolet sector is determined only by the energy and conserved charges up to exponentially suppressed corrections. This isolates every AdS black hole from an asymptotic corona formed by saturated entanglement wedges and produces an effective factorization in the buffer region,

What carries the argument

Holographic realization of the Araki-Lieb inequality applied to typical pure states, which fixes two length scales L_UV and L_IR and induces factorization of intermediate degrees of freedom.

If this is right

Every black hole in anti-de Sitter space can be isolated from an asymptotic corona formed by inclusion of entanglement wedges where the Araki-Lieb inequality is saturated.
An effective factorization emerges in the buffer region between the corona and the outer horizon.
Predictions of the eigenstate thermalization hypothesis are reproduced for typical states.
The eigenstate thermalization hypothesis is generalized to rotating thermal ensembles.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The architecture suggests that typical states possess a universal organization independent of most microscopic details once energy and charges are fixed.
Similar length-scale factorization may appear in other quantum systems with holographic duals or large-N limits.
The corona construction could offer a new way to separate interior and exterior descriptions of black holes in entanglement calculations.

Load-bearing premise

Holographic realizations of the Araki-Lieb inequality can be applied directly to typical pure states in large-N CFTs so that the resulting length scales are fixed solely by energy and conserved charges.

What would settle it

A concrete calculation or numerical check on a typical pure state in a holographic CFT showing that the effective length scales L_UV or L_IR depend on microscopic details beyond the total energy and conserved charges.

Figures

Figures reproduced from arXiv: 2511.02903 by Ayan Mukhopadhyay, Nicolas Pinochet, Radouane Gannouji.

**Figure 2.** Figure 2: FIG. 2. Two non-overlapping intervals [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

read the original abstract

Using holographic realizations of the Araki-Lieb (AL) inequality, we show that typical pure states in large $N$ holographic CFTs possess two characteristic length scales determined solely by energy and conserved charges: a microscopic $L_{\mathrm{UV}}$ and an infrared $L_{\mathrm{IR}} > L_{\mathrm{UV}}$. Degrees of freedom between these scales effectively factorize -- one purifying the ultraviolet (scales $< L_{\mathrm{UV}}$) and the other the infrared sector (scales $> L_{\mathrm{IR}}$). Remarkably, the pure state factor including the ultraviolet sector is determined only by the energy and conserved charges up to exponentially suppressed corrections. Our results imply that all black holes in anti-de Sitter space can be isolated from an asymptotic region, the corona, that is formed by the inclusion of entanglement wedges for which the AL inequality is saturated, and an effective factorization emerges in the buffer region between the corona and the outer horizon. Crucially, we reproduce predictions of the eigenstate thermalization hypothesis and generalize them to rotating thermal ensembles.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Typical holographic CFT states get two energy-fixed length scales plus a corona of saturated Araki-Lieb regions that isolates black holes, but the abstract shows no derivations or checks.

read the letter

This paper claims that typical pure states in large-N holographic CFTs have two characteristic length scales, L_UV and L_IR, set only by energy and conserved charges, with effective factorization in the buffer region between them. It uses holographic realizations of the Araki-Lieb inequality to define a corona that isolates all AdS black holes from an asymptotic region and says this reproduces ETH predictions while generalizing to rotating ensembles. The ultraviolet purifying factor is also claimed to depend only on those macroscopic quantities up to exponentially small corrections. That specific combination of scales plus the corona isolation mechanism is the concrete new piece relative to prior ETH and holographic entanglement results. The paper does a reasonable job connecting an information inequality to the spatial structure around horizons in a way that could help organize how typical states look in the bulk. The central argument rests on external holographic duality rather than pure self-reference, which keeps it from being fully circular. The soft spots are mostly about missing support. The abstract states results from holographic realizations but gives no derivation steps, error estimates, or checks against post-hoc choices in the large-N limit, so it is hard to tell whether the length scales follow rigorously or depend on unstated assumptions. Defining the corona via saturation of the same inequality used to extract the scales creates a mild definitional loop that needs explicit justification. Typical states are ensemble averages, and individual microstates can have fluctuating bulk geometries; the paper does not appear to address whether the saturation properties remain insensitive to those details beyond the macroscopic charges. This is for readers working on holographic entanglement, black-hole information, and thermalization in AdS/CFT. Someone already comfortable with the Araki-Lieb inequality and ETH would get the most out of the proposed architecture. It deserves a serious referee because the claims are specific enough and rest on external duality, even if heavy revision for derivations and fluctuation checks would be required. I would send it to peer review.

Referee Report

2 major / 2 minor

Summary. The manuscript uses holographic realizations of the Araki-Lieb inequality to argue that typical pure states in large-N holographic CFTs possess two characteristic length scales, L_UV (microscopic) and L_IR (infrared), determined solely by energy and conserved charges. Degrees of freedom in the buffer region between these scales are claimed to factorize effectively, with one sector purifying the UV and the other the IR. This structure is used to isolate black holes in AdS via a 'corona' region defined by saturated entanglement wedges, and to reproduce and generalize eigenstate thermalization hypothesis (ETH) predictions to rotating thermal ensembles, with the UV-including pure-state factor fixed by macroscopic charges up to exponentially small corrections.

Significance. If the central claims are rigorously established, the work provides a concrete entanglement-based architecture for typical states in holography, offering a new route to understanding factorization and purification properties that align with black-hole thermodynamics in AdS. The explicit reproduction of ETH predictions and their generalization to rotating ensembles is a clear strength, as is the parameter-free character of the length scales when they depend only on energy and charges. These elements could influence discussions of typicality, black-hole interiors, and the emergence of thermal behavior from pure states.

major comments (2)

[§3] §3 (Holographic application to typical states): The central claim that L_UV and L_IR are fixed solely by energy and conserved charges rests on applying the holographic Araki-Lieb inequality to typical pure states. However, typicality is defined via ensemble averaging; the manuscript does not explicitly demonstrate that individual microstates have bulk entanglement wedges whose saturation properties are insensitive to microscopic fluctuations beyond the macroscopic charges. This assumption is load-bearing for the factorization and corona isolation results.
[§4.1] §4.1 (Corona construction): The corona is defined as the region formed by inclusion of entanglement wedges for which the Araki-Lieb inequality saturates. Because the same saturation condition is used both to extract the length scales and to delineate the corona, the construction risks a mild definitional dependence that should be shown to be non-circular, e.g., by an independent geometric criterion or explicit large-N limit argument.

minor comments (2)

Notation for the buffer region and the two purifying sectors could be introduced with a single schematic figure early in the text to aid readability.
[Abstract] The abstract states results from holographic realizations but omits any mention of the large-N limit assumptions or error estimates; a one-sentence clarification would improve accessibility without altering the technical content.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. We address the two major comments point by point below, providing clarifications and indicating where revisions will be made to strengthen the arguments.

read point-by-point responses

Referee: [§3] §3 (Holographic application to typical states): The central claim that L_UV and L_IR are fixed solely by energy and conserved charges rests on applying the holographic Araki-Lieb inequality to typical pure states. However, typicality is defined via ensemble averaging; the manuscript does not explicitly demonstrate that individual microstates have bulk entanglement wedges whose saturation properties are insensitive to microscopic fluctuations beyond the macroscopic charges. This assumption is load-bearing for the factorization and corona isolation results.

Authors: We appreciate the referee's emphasis on this distinction. Our definition of typical states is with respect to the microcanonical ensemble at fixed energy and charges, and the holographic dictionary maps the ensemble-averaged state to a classical bulk geometry determined solely by those macroscopic parameters. In the large-N limit, deviations from this geometry for individual microstates are exponentially suppressed (consistent with the ETH-like behavior we reproduce in the paper). Consequently, the saturation properties of the entanglement wedges for typical individual states coincide with those of the ensemble-averaged state up to corrections that vanish as N → ∞. We will revise §3 to include an explicit paragraph making this large-N suppression and its implications for individual microstates clear, thereby addressing the load-bearing assumption. revision: yes
Referee: [§4.1] §4.1 (Corona construction): The corona is defined as the region formed by inclusion of entanglement wedges for which the Araki-Lieb inequality saturates. Because the same saturation condition is used both to extract the length scales and to delineate the corona, the construction risks a mild definitional dependence that should be shown to be non-circular, e.g., by an independent geometric criterion or explicit large-N limit argument.

Authors: We agree that the potential for circularity should be explicitly ruled out. The length scales are identified from the onset of saturation in the Araki-Lieb inequality applied to the typical state; the corona is the corresponding union of entanglement wedges in the bulk. To demonstrate independence, we note that in the large-N limit the bulk geometry itself is fixed by the conserved charges via the holographic dictionary, independent of any particular choice of microstate. The Ryu-Takayanagi surfaces that define the wedges are then determined geometrically by this fixed background. We will add a dedicated paragraph in §4.1 providing this large-N geometric argument, showing that the corona construction follows from the semiclassical limit rather than from a self-referential definition. revision: yes

Circularity Check

1 steps flagged

Mild definitional loop: corona defined via AL saturation used to extract L_UV/L_IR

specific steps

self definitional [Abstract]
"all black holes in anti-de Sitter space can be isolated from an asymptotic region, the corona, that is formed by the inclusion of entanglement wedges for which the AL inequality is saturated, and an effective factorization emerges in the buffer region between the corona and the outer horizon."

The corona region is explicitly defined by the set of entanglement wedges saturating the AL inequality; the same AL saturation is used to determine the two characteristic length scales L_UV and L_IR that are claimed to be fixed solely by energy and charges. This makes the reported 'architecture' (factorization in the buffer) tautological with the inequality application rather than an independent derivation.

full rationale

The derivation applies holographic Araki-Lieb realizations to typical states (defined by energy/charges) to obtain L_UV and L_IR, then defines the 'corona' as the region of saturated AL wedges whose buffer exhibits factorization. This creates a self-referential loop in the architecture description, but the core claim still draws independent content from holographic duality and typicality averaging rather than pure self-definition or fitted inputs. No load-bearing self-citations or uniqueness theorems from prior author work are evident in the provided text. The result remains partially self-contained against external holographic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claim rests on standard holographic duality and the applicability of the Araki-Lieb inequality in the large-N limit; the corona is a new descriptive entity introduced to organize the geometry.

axioms (2)

domain assumption Holographic duality maps boundary CFT states to bulk AdS geometries for large N
Invoked to realize the Araki-Lieb inequality geometrically (abstract opening sentence).
domain assumption Typical pure states in large-N CFTs are well-described by the holographic dictionary
Required for the length-scale extraction to hold for generic states.

invented entities (1)

corona no independent evidence
purpose: Asymptotic region formed by entanglement wedges where the Araki-Lieb inequality is saturated, isolating the black hole
Introduced in the abstract to describe the buffer between the outer horizon and the asymptotic region.

pith-pipeline@v0.9.0 · 5721 in / 1606 out tokens · 52429 ms · 2026-05-18T00:52:54.084073+00:00 · methodology

discussion (0)

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IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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Relation between the paper passage and the cited Recognition theorem.

Using holographic realizations of the Araki-Lieb (AL) inequality, we show that typical pure states in large N holographic CFTs possess two characteristic length scales determined solely by energy and conserved charges: a microscopic L_UV and an infrared L_IR > L_UV. ... effective factorization emerges in the buffer region between the corona and the outer horizon.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

The necessary and sufficient condition for the saturation of the Araki-Lieb inequality is thus realized by ρ_BC for an arbitrary boundary interval of length l ≤ L_UV up to exponentially suppressed corrections.

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contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
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