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Minimax surfaces and the holographic entropy cone

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper proves that a stable minimax surface is exactly the HRT surface, and that cooperating time-sheets would yield a graph model making the static and time-dependent holographic entropy cones coincide.

desk verdict Genuinely useful minimax properties plus a clearly conditional graph-model construction; the central theorem leans on an assumed, unproven existence claim, but the paper is honest about it and deserves peer review. read the letter →

arxiv 2502.09894 v2 pith:H7S6DK6V submitted 2025-02-14 hep-th gr-qc

Brianna Grado-White , Guglielmo Grimaldi , Matthew Headrick , Veronika E. Hubeny This is my paper
classification hep-thgr-qc
keywords holographicentanglemententropyminimaxsurfacesHRTformulatime-sheetswedgeconegraphmodelsnullenergycondition
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

The paper establishes that the minimax prescription for holographic entanglement entropy—maximize the area of a surface on a timelike "time-sheet," then minimize over time-sheets—produces exactly the same surface and area as the covariant HRT formula. It proves that a stable minimax surface is an HRT surface (Theorem 2.6), and that the entanglement wedge is the smallest spacetime homology region bounded by a minimax time-sheet. The paper then shows that if a collection of time-sheets "cooperates"—every partial HRT surface is maximal on its partial time-sheet—the time-sheets define a graph on which min cuts compute HRT entropies. That would imply the time-dependent (HRT) entropy cone equals the static (RT) cone; the cooperating property is proved for pairs but only conjectured for arbitrary collections, with explicit examples where it may fail.

What carries the argument

Time-sheets are piecewise timelike or null hypersurfaces homologous to $D(A)$ relative to the future and past conformal boundaries; the minimax surface is the maximal-area surface on a time-sheet, minimized over time-sheets. The load-bearing mechanism is the null focusing theorem (null energy condition plus Einstein equations): focusing makes HRT surfaces maximal on their entanglement horizons, so the infimum over time-sheets selects the least-area extremal surface. The cooperating property—each partial HRT surface is maximal on each partial time-sheet cut out by intersections—is the mechanism that turns a collection of time-sheets into a graph model: graph cuts correspond to unions of partial time-sheets, and cooperation prevents shortcut cuts from undercutting the true entropies.

What would settle it

A concrete check would be to exhibit an asymptotically AdS spacetime satisfying the null energy condition in which no smooth everywhere-timelike minimax time-sheet exists, or where a stable minimax surface has nonzero extrinsic curvature trace; either would break Theorem 2.6. For the cooperating conjecture, an explicit three-region configuration (for instance, a sharpened version of the "small triangle" setup in pure AdS$_3$) with a proof that no cooperating time-sheet configuration exists would falsify Conjecture 4.1 while leaving the cone-equality question open.

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Extended reading notes

Core claim

The central claim is that the minimax formula $S(A)=\frac{1}{4G_N}\inf_{\tau}\sup_{\gamma\in\Gamma_\tau}|\gamma|$ — maximize area over surfaces $\gamma$ on a time-sheet $\tau$ homologous to the boundary domain of dependence $D(A)$, then minimize over time-sheets — computes the same entropy as the covariant HRT prescription. The paper proves that a stable minimax surface is an HRT surface: stability and extremality coincide for minimax surfaces, and the minimization over time-sheets selects the least-area extremal surface in the spacetime homology class. It also proves that the entanglement wedge is the smallest homology region bounded by a minimax time-sheet, that causal holographic information upper-bounds $S(A)$, and that entanglement wedges nest for nested boundary domains of dependence. For crossing regions, cooperating time-sheet configurations are introduced; a cooperating pair always exists for two crossing regions with connected time-sheets. For more regions the paper leaves the existence of cooperating configurations as a conjecture, noting that its graph-model and entropy-cone conclusions are conditional on it.

Load-bearing premise

The proofs assume that a smooth, everywhere timelike minimax time-sheet exists and that the supremum and infimum in the minimax formula are achieved; the graph-model conclusion additionally assumes that cooperating configurations of time-sheets exist for arbitrarily many regions.

Editorial extensions

If this is right

  • A stable minimax surface exists and equals the HRT surface, so holographic entanglement entropy can be computed without first assuming an extremal surface exists.
  • The entanglement wedge is the smallest spacetime homology region bounded by a minimax time-sheet, sharpening the role of the wedge in bulk reconstruction.
  • For any pair of crossing boundary regions, a cooperating pair of time-sheets exists, yielding a minimax-based proof of strong subadditivity that bypasses the maximin representative trick.
  • If the cooperating conjecture holds for all $N$, every time-dependent holographic state with $N$ specified regions admits a weighted graph whose min cuts give all HRT entropies; hence the HRT entropy cone coincides with the RT cone.
  • The relaxed minimax prescription—maximizing over surfaces achronal only within each time-sheet—agrees with the original minimax under the null energy condition, providing an equivalent formulation for entropy-inequality proofs.

Reading between the lines

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

  • If the cooperating conjecture fails for three or more time-sheets, the graph-model route to cone equality is blocked, but cone equality could still hold: the paper's potential counterexample lives in a 2+1-dimensional spacetime where the RT inequalities are already known to hold.
  • A more flexible graph construction—for example allowing non-minimally intersecting configurations or using relaxed-minimax surfaces—might rescue a graph model even where strict cooperation seems impossible.
  • The minimax formulation suggests a quantum version, $S(A)=\min_\tau\max_\sigma(|\gamma|+S(\rho_{\tau\cap\sigma}))$, which could extend the approach to quantum extremal surfaces and bulk entropy inequalities; the paper raises this as a direction rather than proving it.
  • If a spacetime graph model exists, tensor-network constructions for static states could carry over to time-dependent states, effectively integrating out bulk time inside the Wheeler-DeWitt patch; the paper leaves this as an application.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. This paper develops the minimax reformulation of the HRT holographic entanglement entropy formula introduced in [23]. The authors define time-sheets and the minimax area (Eq. (2.2)), prove that a stable minimax surface is extremal (Corollary 2.3, Theorem 2.5) and hence an HRT surface (Theorem 2.6), establish the equivalence between the original and relaxed minimax prescriptions (Lemma 2.7, Theorem 2.8, Corollary 2.9), and prove several geometric properties: the entanglement wedge is the smallest minimax homology region (Theorem 3.1), causal holographic information upper-bounds entanglement entropy (Theorem 3.2), entanglement wedge nesting and spacelike separation of degenerate minimax surfaces hold (Theorem 3.3 and Corollary 3.4), and cooperating pairs of time-sheets exist for connected crossing regions (Theorem 3.5). On this basis the authors construct a candidate spacetime graph model and prove that, if a configuration of cooperating time-sheets exists, graph cuts reproduce HRT entropies (Theorem 4.1), which would imply equality of the RT and HRT entropy cones. They present analytic and numerical evidence for the cooperating conjecture in pure AdS3 and in spherically symmetric matter, and discuss a configuration in pure AdS3 for which cooperation appears difficult.

Significance. If the main theorems hold, minimax provides an independent covariant route to HRT and a spacetime-level tool for proving entropy inequalities, and the graph model would be a substantial step toward showing that the HRT and RT entropy cones coincide. The paper's strengths include the new relaxed-minimax equivalence, the direct proof that the entanglement wedge is the smallest minimax homology region, and a transparent numerical test of the cooperating conjecture. The authors also honestly flag the main unresolved points: the existence of smooth timelike minimax time-sheets, the disconnected-component case of Theorem 3.5, and the open status of the cooperating conjecture. Because the central results are conditional on these points, the contribution is significant but not yet fully established.

major comments (4)
  1. [2.2, Eq. (2.2)] The variational problem (2.2) is the foundation of the paper, but the text immediately following it assumes that the sup and inf are achieved and then states: "we make the assumption that one such everywhere timelike and smooth time-sheet exists." These existence assumptions are load-bearing: Lemma 2.2(i) invokes the smooth timelike time-sheet to conclude smoothness of the minimax surface, and Corollary 2.3 and Theorem 2.6 use that conclusion. No proof or sufficient condition is given, and the "floppiness" argument is explicitly deferred. Please either prove existence under the stated holographic assumptions (NEC, Einstein equations, AdS boundary conditions), or state Theorem 2.6 and its corollaries as conditional theorems with the assumptions made explicit.
  2. [Lemma 2.2(i)] Even granting the existence assumption, the treatment of non-smooth time-sheets is not justified. The proof asserts that if the maximal surface on a non-smooth time-sheet inherits a kink, "a nearby time-sheet which resolves the kink would have a smaller-area maximal surface." This is a local variational statement, while minimax minimization is global over all time-sheets, so the assertion does not follow from minimality of the time-sheet. A rigorous argument is needed to rule out kinks on minimizing time-sheets; as written, the smoothness of the minimax surface is assumed rather than derived.
  3. [Theorem 3.5] The theorem claims existence of cooperating time-sheet pairs for crossing regions, but the proof is a sketch and explicitly leaves out the case of disconnected minimax surfaces: "we suspect that it should go through by applying the same procedure to each connected component individually." The proof also assumes without discussion that the minimum over configurations in (3.3) is attained (footnote 10). Since disconnected components are needed for the proof of SSA and for the general graph-model construction, this is a genuine gap in a central claim. Please either supply the full proof or clearly mark Theorem 3.5 as proven only for connected minimax time-sheets.
  4. [Theorem 3.6] In the equality case |γ+(α)_1| = |γ+(β)_2|, the proof says one can "round off these corners" to obtain a smaller-area maximal surface, but no construction is given and it is not shown that the rounded time-sheet remains in the correct homology class and remains a minimax time-sheet. Since this lemma is used to restrict cooperating configurations to minimally intersecting ones, it needs a rigorous proof or an explicit weakening.
minor comments (5)
  1. [2.1] There is a typo, "codmiension", which should be "codimension".
  2. [2.3, Lemma 2.1] The term "non-degeneracy" is used in the proof but not defined; the argument that a small deformation produces a maximal surface arbitrarily close in area to the original one would benefit from a precise statement of the topology on the space of surfaces.
  3. [4.3] The notation "kmax = 8/13ϵ" is unclear: it would help to specify the normalization of the affine parameter and to state explicitly that the subsequent time interval is of order ϵ.
  4. [4.1] In the graph-model definition, the weight of an edge is described as the area of the partial minimax surface on the shared partial time-sheet; if that partial surface has several connected components, the text should state explicitly that the weight is the sum of their areas.
  5. [4.2.1] The numerical statement that the second eigenvalue of the Hessian changes sign precisely at χ = χc is presented without an analytic expression for the eigenvalue, which makes the check harder to reproduce independently.

Circularity Check

0 steps flagged · score 2.0 of 10

No circularity found: minimax=HRT is argued independently of self-cited results; remaining load-bearing assumptions are explicit existence gaps, not definitional loops.

full rationale

The paper's central theorem (Theorem 2.6) does not reduce to its inputs. The minimax prescription (2.2) is taken from [23], a self-citation, but the paper redefines the objects and supplies a stability-based proof that a stable minimax surface is extremal (Corollary 2.3) and then least-area among extremal surfaces (Theorem 2.6). The alternative argument in Section 2.2 that invokes [23] is explicitly flagged as a reiteration, and the paper notes the second route does not use maximin. The proofs of relaxed-minimax equivalence (Theorem 2.8), entanglement-wedge minimality (Theorem 3.1), and cooperating-pair existence (Theorem 3.5) are carried out with focusing arguments and contradiction constructions rather than by citing the conclusion. The self-citations present ([23] for the formulation, [8] for a technical disjointness corollary in Lemma 2.4) are transparent and not load-bearing for the main equality. The paper explicitly assumes existence of minimax time-sheets and of a smooth everywhere-timelike minimax time-sheet (Section 2.2: 'we make the assumption that one such everywhere timelike and smooth time-sheet exists'), and Lemma 2.2(i) invokes this assumption. This is a genuine unproven-existence gap that would undercut Theorem 2.6 if it fails, but it is not a circular reduction: the assumption is not definitionally equivalent to the HRT statement. The cooperating conjecture and spacetime graph model are presented as conjectural, with explicit acknowledgement of possible failure. Overall, no step in the derivation chain is equivalent by construction to its own inputs; score 2 reflects only the presence of minor, non-load-bearing self-citations and the explicit existence assumptions.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claims rest on standard holographic assumptions (NEC, Einstein equations, AdS) plus two unproven existence assumptions about minimax time-sheets. No free parameters are fitted to data; parameters in the examples (delta, L, E) are inputs defining the spacetimes, not fitted constants.

assumptions (5)
  • domain assumption Null energy condition: T_ab k^a k^b >= 0 for all null k^a
    Invoked throughout the proofs via focusing arguments (section 2, 'we will also assume matter to satisfy the null energy condition').
  • domain assumption Generic condition: on any null geodesic segment there is a point where R_ab k^a k^b != 0
    Ensures strict focusing of null congruences (section 2).
  • ad hoc to paper Existence of minimax time-sheets and surfaces achieving the infimum and supremum
    Assumed in section 2.2 ('We will assume that the sup and inf are achieved'); not proven, and load-bearing for all theorems.
  • ad hoc to paper Existence of a smooth, everywhere timelike minimax time-sheet
    Assumed in section 2.2 ('we make the assumption that one such everywhere timelike and smooth time-sheet exists').
  • domain assumption Einstein field equations and asymptotically AdS boundary conditions
    Standard holographic assumptions used to relate bulk geometry to boundary entanglement (section 2).

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Pith. "Pith review of Minimax surfaces and the holographic entropy cone." pith.science (2026). https://pith.science/paper/H7S6DK6V

@misc{pith2026250209894,
  author       = {Pith},
  title        = {Pith review of: Minimax surfaces and the holographic entropy cone},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/H7S6DK6V}},
  note         = {Machine review of arXiv:2502.09894}
}
read the original abstract

We study and prove properties of the minimax formulation of the HRT holographic entanglement entropy formula, which involves finding the maximal-area surface on a timelike hypersurface, or time-sheet, and then minimizing over the choice of time-sheet. In this formulation, the homology condition is imposed at the level of the spacetime: the homology regions are spacetime volumes rather spatial regions. We show in particular that the smallest minimax homology region is the entanglement wedge. The minimax prescription suggests a way to construct a graph model for time-dependent states, a weighted graph on which min cuts compute HRT entropies. The existence of a graph model would imply that HRT entropies obey the same inequalities as RT entropies, in other words that the RT and HRT entropy cones coincide. Our construction of a graph model relies on the time-sheets obeying a certain ``cooperating'' property, which we show holds in some examples and for which we give a partial proof; however, we also find scenarios where it may fail.

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. The Holographic Multi-Entropy Cone

    hep-th 2026-06 accept novelty 7.0 of 10

    Holographic multi-entropy vectors form a rational polyhedral cone; its n=3,4 facets yield seven fundamental multi-entropy inequality orbits, with ordinary HEC facets arising as convex combinations of HMEC facets.

  2. On the construction of graph models realizing given entropy vectors

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    An efficient algorithm constructs candidate simple tree graph models for entropy vectors that pass a chordality test, but its correctness remains conjectural.

Reference graph

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