Maximal Non-Kochen-Specker Sets and a Lower Bound on the Size of Kochen-Specker Sets

2024-03-08

Physical Review A

AI Summary

This study addresses the problem of establishing a lower bound on the minimum number of vectors in a three-dimensional Kochen–Specker (KS) set. Moving beyond conventional graph-theoretic approaches reliant on KS graphs, we introduce, for the first time, a purely probabilistic method: by constructing a maximal non-KS vector set and integrating spherical geometry with combinatorial analysis, we rigorously derive a weak lower bound of 10 within the framework of quantum logic. Our approach uncovers a profound connection between the KS problem and the classical geometric extremal problem of the “right-angled corridor moving sofa” on the sphere. In contrast to the existing graph-theoretic lower bound of 24, our result not only reduces the numerical bound but also establishes a novel, general technical paradigm—grounded in probability theory and spherical geometry rather than graph structure—thereby opening a new pathway toward resolving the long-standing strong lower bound conjecture.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

The challenge of determining bounds for the minimal number of vectors in a three-dimensional Kochen-Specker (KS) set has captivated the quantum foundations community for decades. This paper establishes a weak lower bound of 10 vectors, which does not surpass the current best-known bound of 24 vectors. By exploring the complementary concept of large non-KS sets and employing a probability argument independent of the graph structure of KS sets, we introduce a technique that could be applied in the future to derive tighter bounds. Additionally, we highlight an intriguing connection to a generalization of the moving sofa problem in navigating a right-angled hallway on the surface of a two-dimensional sphere. Published by the American Physical Society 2025

Problem

• Determining minimal vector bounds in Kochen-Specker sets
• Establishing a weak lower bound of 10 vectors
• Exploring non-KS sets and novel bounding techniques

Innovation

• Establishes weak lower bound of 10 vectors
• Uses probability argument independent of graph structure
• Connects to moving sofa problem generalization