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arXiv:0906.2693 · Journal of Physics A Mathematical and General

Non-bicolourable finite configurations of rays and their deformations

Artur E Ruuge

Published Feb 22, 2006• 1 citation
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Abstract

A new infinite family of examples of finite non-bicolourable configurations of rays in Hilbert space is described. Such configurations appear in the analysis of quantum mechanics in terms of Bell's inequalities and the 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 4 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.

DOI
10.1088/0305-4470/39/10/014

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A new infinite family of examples of finite non-bicolourable configurations of rays in Hilbert space is described. Such configurations appear in the analysis of quantum mechanics in terms of Bell's inequalities and the 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 4 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.
The Kochen‐Specker theorem in quantum mechanics motivates the following combinatorial problem: how to construct a finite orthoalgebra which does not admit a two‐valued probability measure? For example, the so‐called “Penrose’s dodecahedron” (a projective configuration in C4) generates such an orthoalgebra. In this report one describes a new infinite family of examples of atomic coherent orthoalgebras with the mentioned properties which is intimately related to the geometry of the group’s E8. The important features of the construction are the following: (1) the atomic elements are naturally indexed by the elements of a disjoint union of linear manifolds of codimension 1 of an N‐dimensional vector space over F2; (2) the description of the orthogonality relation involves a pair of Z/2Z‐valued functions on Z/4Z; (3) the symmetry of the construction is described in terms of an extension of GL(N, F2); (4) the whole construction works only if N is divisible by 4 (a phenomenon of periodicity).
Abstract. The Kochen-Specker theorem states that a 3-dimensional complex Euclidean space admits a finite configuration of projective lines such that the corresponding quantum observables (the orthogonal projectors) cannot be assigned with 0 and 1 values in a classically consistent way. This paper shows that the irreducible root systems of exceptional and of non-crystallographic types are useful in constructing such configurations in other dimensions. The cases E6 and E7 lead to new examples, while F4, E8, and H4, yield a new interpretation of the known ones. The described configurations have an additional property: they are saturated, i.e. the tuples of mutually orthogonal lines, being partially ordered by inclusion, yield a poset with all maximal elements having the same cardinality (the dimension of space). 1.
The Kochen-Specker theorem states that a 3-dimensional complex Euclidean space admits a finite configuration of projective lines such that the corresponding quantum observables (the orthogonal projectors) cannot be assigned with 0 and 1 values in a classically consistent way. This paper shows that the irreducible root systems of exceptional and of non-crystallographic types are useful in constructing such configurations in other dimensions. The cases E6E_6E6​ and E7E_7E7​ lead to new examples, while F4F_4F4​, E8E_8E8​, and H4H_4H4​, yield a new interpretation of the known ones. The described configurations have an additional property: they are saturated, i.e. the tuples of mutually orthogonal lines, being partially ordered by inclusion, yield a poset with all maximal elements having the same cardinality (the dimension of space).
The Bell - Kochen - Specker theorem against non-contextual hidden variables can be proved by constructing a finite set of `totally non-colourable' directions, as Kochen and Specker did in a Hilbert space of dimension n = 3. We generalize Kochen and Specker's set to Hilbert spaces of any finite dimension , in a three-step process that shows the relationship between different kinds of proofs (`continuum', `probabilistic', `state-specific' and `state-independent') of the Bell - Kochen - Specker theorem. At the same time, this construction of a totally non-colourable set of directions in any dimension explicitly solves the question raised by Zimba and Penrose about the existence of such a set for n = 5.