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Combinatorial properties of holographic entropy inequalities

Guglielmo Grimaldi, Matthew Headrick, Veronika E. Hubeny, Pavel Shteyner

TL;DR

This work develops a comprehensive combinatorial framework for holographic entropy inequalities (HEIs) grounded in the min-cut representation of holographic entropies. It proves a precise equivalence between contraction maps and inclusion dominance for centered inequalities, and shows that null reductions of superbalanced HEIs preserve HEIs while enforcing a majorization constraint on reductions. The authors resolve Grimaldi:2025jad's conjectures by proving two and refuting two, and establish a chain of implications from dominance to region dominance that constrains the holographic entropy cone. The results provide strong evidence that time-dependent holographic states obey the same HEI constraints as static states and offer new tools to analyze the structure and generation of HEIs, with potential implications for the HEC and holographic graph models.

Abstract

A holographic entropy inequality (HEI) is a linear inequality obeyed by Ryu-Takayanagi holographic entanglement entropies, or equivalently by the minimum cut function on weighted graphs. We establish a new combinatorial framework for studying HEIs, and use it to prove several properties they share, including two majorization-related properties as well as a necessary and sufficient condition for an inequality to be an HEI. We thereby resolve all the conjectures presented in [arXiv:2508.21823], proving two of them and disproving the other two. In particular, we show that the null reduction of any superbalanced HEI passes the majorization test defined in [arXiv:2508.21823], thereby providing strong new evidence that all HEIs are obeyed in time-dependent holographic states.

Combinatorial properties of holographic entropy inequalities

TL;DR

This work develops a comprehensive combinatorial framework for holographic entropy inequalities (HEIs) grounded in the min-cut representation of holographic entropies. It proves a precise equivalence between contraction maps and inclusion dominance for centered inequalities, and shows that null reductions of superbalanced HEIs preserve HEIs while enforcing a majorization constraint on reductions. The authors resolve Grimaldi:2025jad's conjectures by proving two and refuting two, and establish a chain of implications from dominance to region dominance that constrains the holographic entropy cone. The results provide strong evidence that time-dependent holographic states obey the same HEI constraints as static states and offer new tools to analyze the structure and generation of HEIs, with potential implications for the HEC and holographic graph models.

Abstract

A holographic entropy inequality (HEI) is a linear inequality obeyed by Ryu-Takayanagi holographic entanglement entropies, or equivalently by the minimum cut function on weighted graphs. We establish a new combinatorial framework for studying HEIs, and use it to prove several properties they share, including two majorization-related properties as well as a necessary and sufficient condition for an inequality to be an HEI. We thereby resolve all the conjectures presented in [arXiv:2508.21823], proving two of them and disproving the other two. In particular, we show that the null reduction of any superbalanced HEI passes the majorization test defined in [arXiv:2508.21823], thereby providing strong new evidence that all HEIs are obeyed in time-dependent holographic states.
Paper Structure (54 sections, 9 theorems, 89 equations, 2 figures)

This paper contains 54 sections, 9 theorems, 89 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Broad overview
  3. Classical and quantum entropies:
  4. Entropies in quantum gravity:
  5. Summary of Grimaldi:2025jad
  6. Results
  7. Note added:
  8. Definitions
  9. Notation & conventions
  10. Min-cut function & entropy vector
  11. Holographic entropy inequalities & the min-cut cone
  12. Matrix notation & contraction maps
  13. Transformations of matrix pairs & inequalities
  14. Balanced & superbalanced inequalities
  15. Centered inequalities & null reduction
  16. ...and 39 more sections

Key Result

Lemma 1

If $(\mathbf{L},\mathbf{R}) \in \mathcal{P}_{\sf N}^{\rm sup}$, then its null reduction $(\mathbf{L}',\mathbf{R}')$ on $i\in[{\sf N}+1]$ is centered on $i$.

Figures (2)

  • Figure 1: Map of the logical implications among the theorems presented in this paper. We organized the map based on the different type of inequalities the theorems apply to: superbalanced ($\mathcal{P}_{\sf N}^{\rm sup}$), centered ($\mathcal{P}_{\sf N}^{\rm cen}$), and balanced ($\mathcal{P}_{\sf N}^{\rm bal}$). We hyperlinked every logical implication to the respective theorem and section, as well as the names in the boxes to their respective definitions. We also show how conjectures \ref{['conj:1']} and \ref{['conj:3']} are resolved (conjectures \ref{['conj:2']} and \ref{['conj:4']} go in the opposite directions, and we provide counterexamples in subsec. \ref{['ssec:dom_PCM']} and \ref{['sssec:converses']} respectively).
  • Figure 2: Examples of star graphs for $k = 3,4,5$ respectively, used in the proof of \ref{['thm:if-balanced-hei-then-region-dominance']}, with the edge weights shown in orange. When $k\neq \mathsf{N}$, all the remaining vertices are isolated and disconnected (and hence omitted from the diagram for visual clarity).

Theorems & Definitions (43)

  • Definition 1: PCM
  • Definition 2: BCM
  • Remark 1
  • Example 1
  • Definition 3: Contraction map
  • Example 2
  • Definition 4: Balance
  • Definition 5: Superbalance
  • Remark 2
  • Example 3
  • ...and 33 more