On Searching for Small Kochen-Specker Vector Systems (extended version)

TL;DR

This paper investigates the minimal size of Kochen-Specker vector systems in R^3, establishing a new lower bound of 18 vectors through theoretical and computational methods, highlighting ongoing gaps in understanding.

Contribution

It introduces a new lower bound for KS vector systems and explores graph-theoretic and topological embedding problems with novel algorithms and extensive experiments.

Findings

Established a lower bound of 18 vectors for KS systems

Developed algorithms for graph-theoretic and topological embedding problems

Identified a significant gap between known bounds for KS system sizes

Abstract

Kochen-Specker (KS) vector systems are sets of vectors in R^3 with the property that it is impossible to assign 0s and 1s to the vectors in such a way that no two orthogonal vectors are assigned 0 and no three mutually orthogonal vectors are assigned 1. The existence of such sets forms the basis of the Kochen-Specker and Free Will theorems. Currently, the smallest known KS vector system contains 31 vectors. In this paper, we establish a lower bound of 18 on the size of any KS vector system. This requires us to consider a mix of graph-theoretic and topological embedding problems, which we investigate both from theoretical and practical angles. We propose several algorithms to tackle these problems and report on extensive experiments. At the time of writing, a large gap remains between the best lower and upper bounds for the minimum size of KS vector systems.