A new characterization of the holographic entropy cone

Guglielmo Grimaldi; Matthew Headrick; Veronika E. Hubeny

arxiv: 2508.21823 · v4 · submitted 2025-08-29 · ✦ hep-th · gr-qc· quant-ph

A new characterization of the holographic entropy cone

Guglielmo Grimaldi , Matthew Headrick , Veronika E. Hubeny This is my paper

Pith reviewed 2026-05-18 19:24 UTC · model grok-4.3

classification ✦ hep-th gr-qcquant-ph

keywords holographic entropy coneRyu-Takayanagi inequalitiesHubeny-Rangamani-Takayanagi formulaMarkov statesmajorization propertyentanglement entropylinear inequalitiescovariant holography

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The pith

A majorization test on Markov states shows that only the RT inequalities define the holographic entropy cone.

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

The paper introduces a test based on Markov states to check which linear inequalities on entanglement entropies must hold for holographic systems. The test simplifies to verifying a majorization property of each candidate inequality. All known Ryu-Takayanagi inequalities satisfy the property, while inequalities outside that set do not. This supplies new evidence that the covariant Hubeny-Rangamani-Takayanagi formula obeys exactly the same inequalities and supplies an independent characterization of the full cone.

Core claim

Using Markov states, we develop a test of this conjecture in a heretofore unexplored regime. The test reduces to checking that a given inequality obeys a certain majorization property, which is easy to evaluate. We find that the RT inequalities pass this test and, surprisingly, only RT inequalities do so. Our results not only provide strong new evidence that the HRT and RT cones coincide, but also offer a completely new characterization of that cone.

What carries the argument

The majorization property check on inequalities when evaluated on Markov states, which filters exactly the Ryu-Takayanagi inequalities from all other linear candidates.

If this is right

The set of valid inequalities for the HRT entropy cone is identical to the set for the RT cone.
Any linear inequality that fails the majorization test on Markov states can be violated by some holographic entropy configuration.
The RT cone admits a characterization as the unique cone whose bounding inequalities all obey the majorization property under Markov-state evaluation.
New candidate inequalities for holographic entropies can now be screened by checking the majorization condition before more involved tests.

Where Pith is reading between the lines

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

The same majorization filter might be applied to other entropy inequalities arising in quantum information to isolate holographic-like constraints.
If the test continues to select only RT inequalities in higher-dimensional or time-dependent settings, it would tighten the link between the two holographic prescriptions.
This approach could be used to generate or rule out candidate inequalities without constructing explicit bulk geometries.

Load-bearing premise

The majorization property test with Markov states is sufficient to decide whether every holographic entropy computed from the HRT formula obeys a given inequality.

What would settle it

A concrete holographic geometry or quantum state in which the HRT formula produces entropies that violate one of the RT inequalities while still satisfying the majorization condition on Markov states.

read the original abstract

Entanglement entropies computed using the holographic Ryu-Takayanagi formula are known to obey an infinite set of linear inequalities, which define the so-called RT entropy cone. The general structure of this cone, or equivalently the set of all valid inequalities, is unknown. It is also unknown whether those same inequalities are also obeyed by entropies computed using the covariant Hubeny-Rangamani-Takayanagi formula, although significant evidence has accumulated that they are. Using Markov states, we develop a test of this conjecture in a heretofore unexplored regime. The test reduces to checking that a given inequality obeys a certain majorization property, which is easy to evaluate. We find that the RT inequalities pass this test and, surprisingly, only RT inequalities do so. Our results not only provide strong new evidence that the HRT and RT cones coincide, but also offer a completely new characterization of that cone.

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

Markov majorization test appears to isolate exactly the RT inequalities for the holographic entropy cone, but abstract-only view leaves the claim unverified.

read the letter

The punchline here is that the authors have developed a test based on a majorization property for Markov states that the RT inequalities pass, and which only those inequalities satisfy according to their check. This gives both a new way to characterize the RT entropy cone and additional support for the idea that the HRT formula obeys exactly the same inequalities. What is new is the application of this majorization test in a regime not previously explored for this purpose. The abstract presents it as reducing to an easy check, which is a practical step forward from listing inequalities or computing in specific cases. They credit the background on Markov states and build directly on the known RT inequalities without claiming to derive them from scratch. The paper does well in highlighting the surprise that no other inequalities pass the test. This could mean the characterization is tight, which would be valuable for understanding the structure of allowed entanglement entropies in holographic theories. The main soft spot is the absence of the full text. Without the derivation of the majorization condition or the specifics of how the test was applied, I cannot verify if it really covers the conjecture across all cases or if there are hidden assumptions in the setup. The claim that this provides strong evidence for HRT-RT coincidence depends on the test being both necessary and sufficient, and that part needs the details to evaluate properly. Since the abstract is all that's here, the soundness remains low for now. This work targets researchers focused on the holographic entropy cone and its implications for quantum gravity. A reader looking for new tools to probe or confirm properties of entanglement in AdS/CFT would find potential value, particularly if the full paper includes explicit examples or proofs. Overall, I recommend sending this to peer review. The central claim is significant for the field if it holds, and the approach is distinct enough that referees should have a chance to examine the full argument and any supporting calculations.

Referee Report

0 major / 1 minor

Summary. The paper develops a test for the conjecture that the Ryu-Takayanagi (RT) and Hubeny-Rangamani-Takayanagi (HRT) holographic entropy cones coincide, based on a majorization property derived from Markov states. The authors report that the known RT inequalities satisfy this test while other candidate inequalities do not, yielding both new evidence for cone coincidence and an alternative characterization of the RT cone.

Significance. If substantiated, the result would strengthen the case for HRT-RT equivalence in a regime not previously probed and supply a practical majorization criterion for validating inequalities. This could streamline future work on the structure of the entropy cone and related questions in holographic entanglement.

minor comments (1)

The abstract asserts that the test was performed and that only RT inequalities pass, but provides no explicit statement of the majorization condition or the precise regime of application.

Simulated Author's Rebuttal

0 responses · 1 unresolved

We thank the referee for their review of our manuscript and for accurately summarizing its main results and potential significance. We note that the major comments section of the report is empty, so there are no specific technical points requiring detailed rebuttal or revision at this stage.

standing simulated objections not resolved

The recommendation is listed as 'uncertain' without any accompanying explanation, specific concerns, or elaboration in the major comments section, so we are unable to address the basis for this assessment.

Circularity Check

0 steps flagged

No circularity detected from available abstract

full rationale

The abstract presents a majorization test on Markov states as an independent probe of the HRT-RT conjecture in a new regime, with the finding that only RT inequalities satisfy the property. No equations, explicit derivations, self-citations, or load-bearing steps are provided in the text, preventing any identification of reductions by construction, fitted inputs renamed as predictions, or ansatzes smuggled via prior work. The central claim is framed as a new characterization derived from the test rather than presupposed by it, rendering the available material self-contained against external benchmarks with no detectable circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard properties of Markov states and majorization from quantum information theory without introducing new fitted parameters or postulated entities.

axioms (1)

domain assumption Markov states possess conditional independence properties that translate into a majorization condition for entropy inequalities
This property is invoked to develop the test in the unexplored regime.

pith-pipeline@v0.9.0 · 5662 in / 1272 out tokens · 61900 ms · 2026-05-18T19:24:21.077605+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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

Using Markov states, we develop a test of this conjecture in a heretofore unexplored regime. The test reduces to checking that a given inequality obeys a certain majorization property
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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

We find that the RT inequalities pass this test and, surprisingly, only RT inequalities do so

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supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
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Forward citations

Cited by 4 Pith papers

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