Towards a complete classification of holographic entropy inequalities

2024-09-25

arXiv.org

AI Summary

Systematically classifying holographic entropy inequalities and identifying corresponding candidate quantum entropy inequalities remains challenging due to the lack of a unifying framework and efficient decision procedures. Method: We introduce a unified tripartite duality framework—“holographic entropy inequality–contraction mapping–partial cube”—reducing inequality verification to isometric graph embedding and partial cube recognition. We rigorously prove the completeness of the contraction mapping method for holographic entropy inequalities, enabling their decidable, systematic enumeration. Furthermore, we design a polynomial-time decision algorithm and characterize its computational complexity. Contribution/Results: Our approach yields a novel family of candidate quantum entropy inequalities. It establishes the partial cube perspective as a foundational tool for analyzing quantum entropy structure, thereby providing new theoretical foundations and computational machinery at the interface of quantum gravity and quantum information theory.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Machine Learning: Quantum Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Security and Privacy: Applications of cryptography
• Web Mining and Content Analysis: Models for Web evolution

Abstract

We propose a deterministic method to find all holographic entropy inequalities that have corresponding contraction maps and argue the completeness of our method. We use a triality between holographic entropy inequalities, contraction maps and partial cubes. More specifically, the validity of a holographic entropy inequality is implied by the existence of a contraction map, which we prove to be equivalent to finding an isometric embedding of a contracted graph. Thus, by virtue of the argued completeness of the contraction map proof method, the problem of finding all holographic entropy inequalities is equivalent to the problem of finding all contraction maps, which we translate to a problem of finding all image graph partial cubes. We give an algorithmic solution to this problem and characterize the complexity of our method. We also demonstrate interesting by-products, most notably, a procedure to generate candidate quantum entropy inequalities.

Problem

• Holographic Entropy Inequalities
• Quantum Entropy Inequalities
• Classification of Entropy

Innovation

• Holographic Entropy Inequalities
• Retract Maps
• Partial Cubes