Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:0906.2687 · Journal of Mathematical Physics

New families of finite coherent orthoalgebras without bivaluations

Artur E. Ruuge, Freddy Van Oystaeyen

Published Feb 1, 2006• 4 citations
arXivHTMLPDF

Abstract

In the present paper we study the following problem: how to construct a coherent orthoalgebra which has only a finite number of elements, but at the same time does not admit a bivaluation (i.e., a morphism with a codomain being an orthoalgebra with just two elements). This problem is important in the perspective of Bell-Kochen-Specker theory, since one can associate such an orthoalgebra to every saturated noncolorable finite configuration of projective lines. The first result obtained in this paper provides a general method for constructing finite orthoalgebras. This method is then applied to obtain a new infinite family of finite coherent orthoalgebras that do not admit bivaluations. The corresponding proof is combinatorial and yields a description of the groups of symmetries for these orthoalgebras.

DOI
10.1063/1.2171691

Related papers

Finite Orthoalgebras without Two-valued Probability Measures

Artur E. Ruuge

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).

AIP conference proceedings•Jan 1, 2007•0 citations

Saturated Kochen–Specker-type configuration of 120 projective lines in eight-dimensional space and its group of symmetry

Artur E. Ruuge, Freddy Van Oystaeyen

There exists an example of a set of 40 projective lines in eight-dimensional Hilbert space producing a Kochen–Specker-type contradiction. This set corresponds to a known no-hidden variables argument due to Mermin. In the present paper it is proved that this set admits a finite saturation, i.e., an extension up to a finite set with the following property: every subset of pairwise orthogonal projective lines has a completion, i.e., is contained in at least one subset of eight pairwise orthogonal projective lines. An explicit description of such an extension consisting of 120 projective lines is given. The idea to saturate the set of projective lines related to Mermin’s example together with the possibility to have a finite saturation allow to find the corresponding group of symmetry. This group is described explicitely and is shown to be generated by reflections. The natural action of the mentioned group on the set of all subsets of pairwise orthogonal projective lines of the mentioned extension is investigated. In particular, the restriction of this action to complete subsets is shown to have only four orbits, which have a natural characterization in terms of the construction of the saturation.

Journal of Mathematical Physics•Apr 27, 2005•14 citations

Non-bicolourable finite configurations of rays and their deformations

Artur E Ruuge

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.

arXiv:0906.2693•Journal of Physics A Mathematical and General•Feb 22, 2006•1 citation

Exceptional and non-crystallographic root systems and the Kochen–Specker theorem

Artur E Ruuge

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 $E_6$ and $E_7$ lead to new examples, while $F_4$, $E_8$, and $H_4$, 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).

arXiv:0906.2696•Journal of Physics A Mathematical and Theoretical•Feb 28, 2007•7 citations

EXCEPTIONAL AND NON-CRYSTALLOGRAPHIC ROOT SYSTEMS AND THE KOCHEN-SPECKER THEOREM

Artur E. Ruuge

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.

•Nov 27, 2012•5 citations

Proofs of the Kochen-Specker theorem based on the N -qubit Pauli group

Mordecai Waegell, P. K. Aravind

We present a number of observables-based proofs of the Kochen-Specker (KS) theorem based on the $N$-qubit Pauli group for $N\ensuremath{\ge}4$, thus adding to the proofs that have been presented earlier for the 2- and 3-qubit groups. These proofs have the attractive feature that they can be presented in the form of diagrams from which they are obvious by inspection. They are also irreducible in the sense that they cannot be reduced to smaller proofs by ignoring some subset of qubits and/or observables in them. A simple algorithm is given for transforming any observables-based KS proof into a large number of projectors-based KS proofs; if the observables-based proof has $O$ observables, with each observable occurring in exactly two commuting sets and any two commuting sets having at most one observable in common, the number of associated projectors-based parity proofs is 2${}^{O}$. We introduce symbols for the observables- and projectors-based KS proofs that capture their important features and also convey a feeling for the enormous variety of both these types of proofs within the $N$-qubit Pauli group. We discuss an infinite family of observables-based proofs, whose members include all numbers of qubits from 2 up, and show how it can be used to generate projectors-based KS proofs involving only 9 bases (or experimental contexts) in any dimension of the form ${2}^{N}$ for $N\ensuremath{\ge}2$. Some implications of our results are discussed.

arXiv:1302.4801•Physical Review A•Jul 1, 2013•33 citations
In the present paper we study the following problem: how to construct a coherent orthoalgebra which has only a finite number of elements, but at the same time does not admit a bivaluation (i.e., a morphism with a codomain being an orthoalgebra with just two elements). This problem is important in the perspective of Bell-Kochen-Specker theory, since one can associate such an orthoalgebra to every saturated noncolorable finite configuration of projective lines. The first result obtained in this paper provides a general method for constructing finite orthoalgebras. This method is then applied to obtain a new infinite family of finite coherent orthoalgebras that do not admit bivaluations. The corresponding proof is combinatorial and yields a description of the groups of symmetries for these orthoalgebras.
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).
There exists an example of a set of 40 projective lines in eight-dimensional Hilbert space producing a Kochen–Specker-type contradiction. This set corresponds to a known no-hidden variables argument due to Mermin. In the present paper it is proved that this set admits a finite saturation, i.e., an extension up to a finite set with the following property: every subset of pairwise orthogonal projective lines has a completion, i.e., is contained in at least one subset of eight pairwise orthogonal projective lines. An explicit description of such an extension consisting of 120 projective lines is given. The idea to saturate the set of projective lines related to Mermin’s example together with the possibility to have a finite saturation allow to find the corresponding group of symmetry. This group is described explicitely and is shown to be generated by reflections. The natural action of the mentioned group on the set of all subsets of pairwise orthogonal projective lines of the mentioned extension is investigated. In particular, the restriction of this action to complete subsets is shown to have only four orbits, which have a natural characterization in terms of the construction of the saturation.
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 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).
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.
We present a number of observables-based proofs of the Kochen-Specker (KS) theorem based on the NNN-qubit Pauli group for $N\ensuremath{\ge}4$, thus adding to the proofs that have been presented earlier for the 2- and 3-qubit groups. These proofs have the attractive feature that they can be presented in the form of diagrams from which they are obvious by inspection. They are also irreducible in the sense that they cannot be reduced to smaller proofs by ignoring some subset of qubits and/or observables in them. A simple algorithm is given for transforming any observables-based KS proof into a large number of projectors-based KS proofs; if the observables-based proof has OOO observables, with each observable occurring in exactly two commuting sets and any two commuting sets having at most one observable in common, the number of associated projectors-based parity proofs is 2O{}^{O}O. We introduce symbols for the observables- and projectors-based KS proofs that capture their important features and also convey a feeling for the enormous variety of both these types of proofs within the NNN-qubit Pauli group. We discuss an infinite family of observables-based proofs, whose members include all numbers of qubits from 2 up, and show how it can be used to generate projectors-based KS proofs involving only 9 bases (or experimental contexts) in any dimension of the form 2N{2}^{N}2N for $N\ensuremath{\ge}2$. Some implications of our results are discussed.