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On the completeness of contraction map proof method for holographic entropy inequalities

Ning Bao, Keiichiro Furuya, Joydeep Naskar

TL;DR

The paper addresses whether the contraction map condition is not only sufficient but also necessary for holographic entropy inequalities (HEIs) in the linear, rational-coefficient case. It develops a geometry-graph duality framework that maps RT arrangements to partial cube graphs and uses this to prove a completeness theorem: an HEI is valid if and only if a contraction map exists. Through simple examples and a rigorous mathematical proof, the authors show that non-contractive maps necessarily induce bulk geometries that violate the RT-based entropy relations, establishing the necessity of contraction maps. The results solidify the contraction map method as a complete algorithmic approach for generating all HEIs within the studied class and illuminate the deep connection between bulk geodesics, entanglement structure, and graph-theoretic representations.

Abstract

The contraction map proof method is the commonly used method to prove holographic entropy inequalities. Existence of a contraction map corresponding to a holographic entropy inequality is a sufficient condition for its validity. But is it also necessary? In this note, we answer that question in affirmative for all linear holographic entropy inequalities with rational coefficients. We show that the pre-image of a non-contraction map is not a hypercube, but a proper cubical subgraph, and show that this manifests as alterations to the geodesic structure in the bulk, which leads to the violation of inequalities by holographic geometries obeying the RT formula.

On the completeness of contraction map proof method for holographic entropy inequalities

TL;DR

The paper addresses whether the contraction map condition is not only sufficient but also necessary for holographic entropy inequalities (HEIs) in the linear, rational-coefficient case. It develops a geometry-graph duality framework that maps RT arrangements to partial cube graphs and uses this to prove a completeness theorem: an HEI is valid if and only if a contraction map exists. Through simple examples and a rigorous mathematical proof, the authors show that non-contractive maps necessarily induce bulk geometries that violate the RT-based entropy relations, establishing the necessity of contraction maps. The results solidify the contraction map method as a complete algorithmic approach for generating all HEIs within the studied class and illuminate the deep connection between bulk geodesics, entanglement structure, and graph-theoretic representations.

Abstract

The contraction map proof method is the commonly used method to prove holographic entropy inequalities. Existence of a contraction map corresponding to a holographic entropy inequality is a sufficient condition for its validity. But is it also necessary? In this note, we answer that question in affirmative for all linear holographic entropy inequalities with rational coefficients. We show that the pre-image of a non-contraction map is not a hypercube, but a proper cubical subgraph, and show that this manifests as alterations to the geodesic structure in the bulk, which leads to the violation of inequalities by holographic geometries obeying the RT formula.

Paper Structure

This paper contains 16 sections, 11 theorems, 91 equations, 7 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Review of the Contraction Map Proof Method
  3. Graphs from holographic geometries and holographic geometries from graphs
  4. Graphs from bitstrings and bitstrings from graphs
  5. Contraction map
  6. Graphs and holographic geometries of HEI from contraction map
  7. Lessons from Simple Scenarios
  8. Deforming the Monogamy of Mutual Information
  9. A Five-Party Example
  10. Non-contraction and Geometry
  11. Main theorem
  12. Mathematical proof
  13. Discussion
  14. Completeness of the algorithmic method to generate all the HEIs
  15. Interpretations of the HEIs
  16. ...and 1 more sections

Key Result

Lemma 2.1

Given an RT arrangement $\Gamma_{J}$, the underlying unit-weighted graph of the RT-region graph $G_J$ is a partial cube whose isometric dimension is $idim(G_J)=J$.

Figures (7)

  • Figure 1: (a) A holographic geometry with the RT arrangement: Four RT surfaces denoted by black lines are arranged on a constant time slice of AdS$_3$/CFT$_2$. The boundary is colored red. (b) The RT arrangement partitions the time slice into bulk subregions. (c) The partial cube associated with the RT arrangement. The boundary vertices are shown in red. The vertex in the middle, shown in black, is a bulk vertex. Its isometric dimension is $4$, which matches the total number of the RT surfaces.
  • Figure 2: The RT surfaces associated with the LHS terms (\ref{['fig:lhs-terms']}) is cut and glued (rearranged) to correspond to (non-minimal) surfaces associated with RHS terms (\ref{['fig:lhs-cut-glue']}) and finally smoothly deformed to their respective minimal surfaces (\ref{['fig:rhs-terms']}).
  • Figure 3: The violations of $d_H \overset{\hbox{adj}}{=} d_G$: The dotted lines emphasize the null edges that violate the adjacency condition. The red circles are the boundary vertices. The black dots are the bulk vertices. (a) The nontrivial preimage $\mathcal{P}(\Phi_{\tilde{f}}(H_3))$ has eight edges missing. Those eight edges belong to the kernel $Ker(\Phi_{\tilde{f}})$. (b) The image graph $\Phi_{\tilde{f}}(H_3)$ corresponding to the non-contraction map $\tilde{f}$ after choosing the empty bits in the last column to be $1,1,0,0$ for the second, third, fifth, and eighth row, respectively.
  • Figure 4: The minimal surface arrangement corresponding to the RHS of inequality \ref{['eq:deformed-mmi']}. It is impossible to smoothly deform the configuration of RT surfaces in figure \ref{['fig:lhs-terms']} into this configuration by cutting and gluing. It requires the RT surface associated with $AB$ to be deformed and not deformed simultaneously.
  • Figure 5: The graph corresponding to the contraction map after choosing the empty bits to be $1,1,0,0$ respectively. This graph (encoding the HEI through occurrence bitstrings) does not have an isometric embedding in $H_5$. The vertices in red correspond to occurrence bitstrings, and those in light-blue are fixed by the deterministic rules described in Bao-2024-properties.
  • ...and 2 more figures

Theorems & Definitions (31)

  • Definition 2.1: RT arrangements in AdS$_{D+1}$/CFT$_D$
  • Remark 1
  • Definition 2.2: RT-region GraphBao:2015bfa
  • Definition 2.3: Partial cubeOvchinnikovWINKLER1984221
  • Definition 2.4: Isometric dimension Ovchinnikov
  • Lemma 2.1
  • Remark 2
  • Definition 2.5: Winkler (equivalence) relation and its equivalence classesOvchinnikovovchinnikov2008partialWINKLER1984221
  • Lemma 2.2: WINKLER1984221ovchinnikov2008partialOvchinnikov
  • Definition 2.6: Semicubesovchinnikov2008partial
  • ...and 21 more