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Covariant Holographic Entropy Cone

Bowen Zhao

TL;DR

The paper addresses whether the covariant holographic entropy cone defined by the HRRT formula can be captured by a graph model, ensuring polyhedrality and a finite set of entropy inequalities for time-dependent spacetimes. It introduces a graph construction from entanglement-wedge boundaries and proves a No-Short-Cut theorem, showing that the discrete entropy $S^*(I)$ equals the covariant HRRT entropy $S(I)$ and that minimization over graph cuts reduces to unions of complete HRT surfaces. This establishes an isomorphism between the covariant and static holographic entropy cones, preserving polyhedrality and finite inequality bases in general dimensions. The work unifies static and covariant perspectives, enabling graph-theoretic and max-flow approaches in dynamic holography and suggesting connections to covariant bit-thread formalisms and broader Lorentzian entanglement structures.

Abstract

The holographic entropy cone classifies the possible entanglement structures of quantum states with a classical gravity dual. For static geometries, Bao et al. established that this cone is polyhedral by constructing a graph model from Ryu-Takayanagi (RT) surfaces on a time-symmetric slice. Extending this framework to general, time-dependent states governed by the Hubeny-Rangamani-Takayanagi (HRT) formula has remained an open problem, as the relevant extremal surfaces do not lie on a common spatial slice. We resolve this by constructing a graph model directly from the causal structure of entanglement wedges. By proving a key "no-short-cut" theorem, we show that minimization over graph cuts reduces to a consideration of cuts corresponding to unions of complete HRT surfaces, establishing the equivalence of the covariant and static holographic entropy cones. Consequently, all foundational results, including polyhedrality and the finite nature of entropy inequalities, extend to general holographic states.

Covariant Holographic Entropy Cone

TL;DR

The paper addresses whether the covariant holographic entropy cone defined by the HRRT formula can be captured by a graph model, ensuring polyhedrality and a finite set of entropy inequalities for time-dependent spacetimes. It introduces a graph construction from entanglement-wedge boundaries and proves a No-Short-Cut theorem, showing that the discrete entropy equals the covariant HRRT entropy and that minimization over graph cuts reduces to unions of complete HRT surfaces. This establishes an isomorphism between the covariant and static holographic entropy cones, preserving polyhedrality and finite inequality bases in general dimensions. The work unifies static and covariant perspectives, enabling graph-theoretic and max-flow approaches in dynamic holography and suggesting connections to covariant bit-thread formalisms and broader Lorentzian entanglement structures.

Abstract

The holographic entropy cone classifies the possible entanglement structures of quantum states with a classical gravity dual. For static geometries, Bao et al. established that this cone is polyhedral by constructing a graph model from Ryu-Takayanagi (RT) surfaces on a time-symmetric slice. Extending this framework to general, time-dependent states governed by the Hubeny-Rangamani-Takayanagi (HRT) formula has remained an open problem, as the relevant extremal surfaces do not lie on a common spatial slice. We resolve this by constructing a graph model directly from the causal structure of entanglement wedges. By proving a key "no-short-cut" theorem, we show that minimization over graph cuts reduces to a consideration of cuts corresponding to unions of complete HRT surfaces, establishing the equivalence of the covariant and static holographic entropy cones. Consequently, all foundational results, including polyhedrality and the finite nature of entropy inequalities, extend to general holographic states.
Paper Structure (17 sections, 7 theorems, 42 equations, 3 figures)

This paper contains 17 sections, 7 theorems, 42 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Notations and Assumptions
  3. Graph models for covariant holographic entropy cone
  4. Definition of the Graph Model
  5. Basic Lemmas
  6. The "No-Short-Cut" Theorem
  7. A worked example of the induction step
  8. Conclusion and Outlook
  9. A Review of Maximum Principles
  10. Maximum Principles and Comparison for the Minimal Surface Operator
  11. Ellipticity and linear operators
  12. Weak and strong maximum principles
  13. The minimal surface operator
  14. Weak comparison principle for $\mathcal{M}$
  15. Strong comparison principle for $\mathcal{M}$
  16. ...and 2 more sections

Key Result

Lemma 2.2

Let $A,B \subset \partial M$ be boundary spatial regions on a common Cauchy slice of the boundary, and let $\gamma_A$ and $\gamma_B$ be their corresponding HRT surfaces in a classical asymptotically AdS spacetime satisfying the null energy condition (NEC) and the usual genericity assumptions. Denote Then $\gamma_A$ can intersect $\partial\mathcal{E}(B)$ at most once along each connected component.

Figures (3)

  • Figure 1: Illustration of no multiple entering of HRT surface into another entanglement wedge. The figure shows the Cauchy slice on which the HRT surface of A (red curve). We also demand $\Sigma \cap \partial M$ contains the spacelike boundary $\partial B$ and $\partial B^*$. The black and green curves denote $\partial \mathcal{E}(B)\cap \Sigma$ and $\partial \mathcal{E}(B^*)\cap \Sigma$, respectively.
  • Figure 2: Illustration of Lemma \ref{['lemma:projection_calculation']} (two intersecting entanglement horizons) in dimension three. In the notation of the lemma, we identify $A = A_1 \cup A_2$ and $B = A_2 \cup A_3$. Accordingly, $\partial\mathcal{E}(A)$ and $\partial\mathcal{E}(B)$ intersect along a seam $\mathcal{S} = \partial\mathcal{E}(A_1A_2)\cap\partial\mathcal{E}(A_2A_3)$. Panel (a) shows the configuration projected onto a single time slice. Panel (b) illustrates the projection of the outer portion $\mathrm{HRT}(A_1A_2)\setminus \mathcal{E}(A_2A_3)$ along null generators onto a common bulk Cauchy slice. Here $\Sigma_A=\Sigma_{12}$ is a maximin slice on which $\mathrm{HRT}(A_1A_2)$ is minimal, while $\Sigma_B=\Sigma_{23}$ is a maximin slice on which both $\mathrm{HRT}(A_2A_3)$ and $\mathrm{HRT}(A_1A_2A_3)$ are minimal. The projection replaces two partial HRT surface pieces by a single complete HRT surface $\mathrm{HRT}(A_1A_2A_3)$ without increasing area. Panel (c) is analogous to Panel (b) and illustrates that the argument does not depend on the relative causal ordering of $p$ and $q$.
  • Figure 3: Illustration of the induction step in the proof of Theorem 2.3 with three intersecting entanglement horizons in dimension three. Panel (a) illustrates the geometric set-up, suppressing the time direction. Panel (b) highlights the change of partial HRT surfaces before and after the projection step. In particular, the union $\mathrm{HRT}(A_2A_3A_4)\cup\mathrm{HRT}(A_3A_4A_5)$ (green and blue) is replaced by $\mathrm{HRT}(A_2A_3A_4A_5)$ (pink). Panel (c) illustrates the projection process along null generators. The thick blue curve $q_4 c$ denotes $\gamma_{345}^{\mathrm{o}}$. The thick orange curve $q_3 p_4$ denotes $\gamma_{234}^{\mathrm{o}}$. The thick purple curve $a p_3$ denotes $\gamma_{123}^{\mathrm{o}}$. Vertical lines indicate pairwise intersections between entanglement horizons. The bulk Cauchy slice $\Sigma_{123}$ is a maximin slice for $\gamma_{123}$, the bulk Cauchy slice $\Sigma_{234}$ is a maximin slice for both $\gamma_{234}$ and $\gamma_{2345}$, and the bulk Cauchy slice $\Sigma_{345}$ is a maximin slice for $\gamma_{345}$.

Theorems & Definitions (14)

  • Definition 2.1: Discrete Entropy
  • Lemma 2.2: No Multi-Crossing of HRT Through Entanglement Horizon
  • proof
  • Theorem 2.3: No-Short-Cut
  • Lemma 2.4: Two intersecting entanglement horizons
  • proof
  • proof
  • Remark 2.5
  • Theorem B.1: Weak maximum principle
  • Theorem B.2: Strong (Hopf) maximum principle
  • ...and 4 more