Non-bicolourable Finite Configurations of Rays and Their Deformations

TL;DR

This paper introduces an infinite family of finite non-bicolorable ray configurations in Hilbert space, relevant to quantum mechanics, and explores their deformations to understand their structural properties.

Contribution

It presents a new class of non-bicolorable configurations parametrized by integers and complex parameters, and introduces a deformation concept to compare these configurations.

Findings

Constructed an infinite family of non-bicolorable configurations.

Defined a deformation notion to compare configurations.

Connected configurations to quantum mechanics principles.

Abstract

A new infinite family of examples of finite non-bicolorable configurations of rays in Hilbert space is described. Such configurations appear in the analysis of quantum mechanics in terms of Bell's inequalities and Kochen-Specker theorem and illustrate that there is no measurable space in the background of the probability model of a quantum system. The mentioned examples are naturally parametrized by a positive integer divisible by four and by several complex-valued parameters, whose number depends on this integer. In order to compare two configurations with the same number of rays, a notion of deformation of a configuration is introduced. The constructed examples are then interpreted as obtained by way of deformations.