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Journal of Mathematical Physics

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

Artur E. Ruuge, Freddy Van Oystaeyen

Published Apr 27, 2005• 14 citations
Publisher

Abstract

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.

DOI
10.1063/1.1887923

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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.
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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 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).
We look at generalisations of sets of vectors proving the Kochen-Specker theorem in 3 and 4 dimensions. It has been shown that two such sets, although unitarily inequivalent, are part of a larger 3-parameter family of vectors that share the same orthogonality graph. We find that these sets are unusual, in that the vectors in all other Kochen-Specker sets investigated here are fully determined by orthog-onality conditions and thus admit no free parameters. The Kochen-Specker (KS) theorem [1] is a statement about the fundamen-tal nature of quantum mechanics: no non-contextual hidden variable theory can reliably reproduce quantum mechanical predictions. The KS theorem states that in a Hilbert space with dimension n ≥ 3 it is impossible to assign definite values v ∈ {0, 1} to m projection operators Pm so that for every commuting set of projection operators satisfying Pm = I, the correspond-ing definite values obey v (Pm) = 1. The KS theorem is often visualised by means of a set of "uncolourable" vectors, where projection operators are represented by their constituent rays in Hilbert space and rays can be simplified to vectors as only directions are relevant here. The vectors are coloured according to their pre-assigned val-ues and the summation conditions above translate into colouring rules. This allows us to construct orthogonality graphs that, when uncolourable, pro-vide a contradiction within non-contextual hidden variable models and thus a proof of the KS theorem. Sets of vectors exemplifying the KS theorem have been discovered by nu-merous authors, with an aim to reduce the number of vectors required to reach a contradiction. Most notably, in a 3-dimensional Hilbert space, the original KS proof of 117 vectors has been replicated with sets using between 31 and 33 vectors [2, 3, 4, 5] and in a 4-dimensional Hilbert space, sets of between 24 and the lower limit of 18 vectors have been realised [2, 6, 7]. Additionally, an extensive computer search has recently revealed over one thousand of such KS sets in R4 [8]. In a publication by E. Gould and P. K. Aravind [10], two of these sets in 3 dimensions have been shown to share the same orthogonality graph. Specifically, the sets of 33 vectors found by Penrose [3] and Peres [2] have
We develop a geometric framework for constructing and organizing Kochen--Specker configurations in real four-dimensional space. The construction uses finite grids on pairs of smooth quadrics related by Euclidean polarity. Their incidence geometry directly produces orthogonal measurement contexts and parity proofs of quantum contextuality, yielding infinite families of configurations and a geometric interpretation and extension of a previously known cyclic construction. For a distinguished subfamily, the same geometry admits a canonical completion determined by secant lines. The first two instances of this completion recover the exceptional root configurations of types F4F_4F4​ and H4H_4H4​, while the smallest case also recovers the Cabello configuration and its embedding in the Peres configuration. Thus several prominent four-dimensional contextual configurations, previously obtained from different constructions, arise from a single projective-geometric mechanism.