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The Holographic Entropy Cone

Ning Bao, Sepehr Nezami, Hirosi Ooguri, Bogdan Stoica, James Sully, Michael Walter

TL;DR

The paper formalizes the holographic entropy cone governing Ryu-Takayanagi entropies for multipartite boundary regions, proving that SSA and MMI fully constrain the cone for up to four regions, and unveiling an infinite family of inequalities for five or more regions. It introduces a finite, graph-based model that exactly reproduces holographic entropies, establishing polyhedrality of the cone and enabling combinatorial proofs via contractions on hypercubes. A new framework of proofs by contraction yields a rich set of higher-order inequalities, including cyclic families, and provides a greedy algorithm to construct such proofs. The results bridge geometry, graph theory, and CFT, linking multiboundary wormhole geometries to entropic building blocks and outlining significant implications for holography, quantum gravity, and tensor-network models.

Abstract

We initiate a systematic enumeration and classification of entropy inequalities satisfied by the Ryu-Takayanagi formula for conformal field theory states with smooth holographic dual geometries. For 2, 3, and 4 regions, we prove that the strong subadditivity and the monogamy of mutual information give the complete set of inequalities. This is in contrast to the situation for generic quantum systems, where a complete set of entropy inequalities is not known for 4 or more regions. We also find an infinite new family of inequalities applicable to 5 or more regions. The set of all holographic entropy inequalities bounds the phase space of Ryu-Takayanagi entropies, defining the holographic entropy cone. We characterize this entropy cone by reducing geometries to minimal graph models that encode the possible cutting and gluing relations of minimal surfaces. We find that, for a fixed number of regions, there are only finitely many independent entropy inequalities. To establish new holographic entropy inequalities, we introduce a combinatorial proof technique that may also be of independent interest in Riemannian geometry and graph theory.

The Holographic Entropy Cone

TL;DR

The paper formalizes the holographic entropy cone governing Ryu-Takayanagi entropies for multipartite boundary regions, proving that SSA and MMI fully constrain the cone for up to four regions, and unveiling an infinite family of inequalities for five or more regions. It introduces a finite, graph-based model that exactly reproduces holographic entropies, establishing polyhedrality of the cone and enabling combinatorial proofs via contractions on hypercubes. A new framework of proofs by contraction yields a rich set of higher-order inequalities, including cyclic families, and provides a greedy algorithm to construct such proofs. The results bridge geometry, graph theory, and CFT, linking multiboundary wormhole geometries to entropic building blocks and outlining significant implications for holography, quantum gravity, and tensor-network models.

Abstract

We initiate a systematic enumeration and classification of entropy inequalities satisfied by the Ryu-Takayanagi formula for conformal field theory states with smooth holographic dual geometries. For 2, 3, and 4 regions, we prove that the strong subadditivity and the monogamy of mutual information give the complete set of inequalities. This is in contrast to the situation for generic quantum systems, where a complete set of entropy inequalities is not known for 4 or more regions. We also find an infinite new family of inequalities applicable to 5 or more regions. The set of all holographic entropy inequalities bounds the phase space of Ryu-Takayanagi entropies, defining the holographic entropy cone. We characterize this entropy cone by reducing geometries to minimal graph models that encode the possible cutting and gluing relations of minimal surfaces. We find that, for a fixed number of regions, there are only finitely many independent entropy inequalities. To establish new holographic entropy inequalities, we introduce a combinatorial proof technique that may also be of independent interest in Riemannian geometry and graph theory.

Paper Structure

This paper contains 39 sections, 13 theorems, 78 equations, 14 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction and summary of results
  2. Organization of the paper.
  3. The holographic entropy cone
  4. Transformations and symmetries
  5. Fewer than five regions
  6. Two regions.
  7. Three regions.
  8. Four regions.
  9. Five and more regions.
  10. Other entropy inequalities.
  11. Uniqueness.
  12. The graph model of holographic entropies
  13. From geometries to graphs
  14. Graph transformations.
  15. From graphs to geometries
  16. ...and 24 more sections

Key Result

Lemma 1

Given a bulk geometry $X$ and boundary regions $A_1, \dots, A_n \subseteq \partial X$, construct the associated graph model as described above. Then $S(I) = S^*(I)$.

Figures (14)

  • Figure 1: The holographic entropy cone ${\mathcal{C}}_2$ for two regions. See \ref{['subsec:few regions']} for a discussion of its facets and extreme rays.
  • Figure 2: Geometries realizing the extreme rays of the holographic entropy cones for $n \leq 4$ regions.
  • Figure 3: The Ryu-Takayanagi surfaces of the $2^n-1$ subsystems cut the bulk geometry into a finite number of pieces.
  • Figure 4: The graph obtained from our construction applied to the bulk geometry and boundary regions in \ref{['fig:tograph-1']}.
  • Figure 5: Graph models of the extreme rays of the holographic entropy cone for $n \leq 4$ regions. All black edges have the same weight; the blue edge has twice that weight.
  • ...and 9 more figures

Theorems & Definitions (26)

  • proof
  • Definition 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 1
  • Lemma 3
  • proof
  • Proposition 1
  • ...and 16 more