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).
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).
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.
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.
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 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).
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
One of the main theorems on the impossibility of hidden variables in quantum mechanics is Kochen-Specker theorem (KS). This theorem says that any hidden variable theory that satisfies quantum mechanics must be contextual. More specifically, it asserts that, in Hilbert space of dimension ≥ 3, it is impossible to associate definite numerical values, 1 or 0, with every projection operator Pm, in such a way that, if a set of commuting Pm satisfies 1=ΣmP, the corresponding values will also satisfy . Since the first proof of Kochen and Specker using 117 vectors in R3, there were many attempts to reduce the number of vector either via conceiving ingenious models or extending the system being considered to higher dimension. By considering eight dimensional three qubits system, we found a state dependent proof that requires only five vectors. The state that we assign value of 1 is the ray that arises from intersection of two planes. The recent advancements show that the KS theorem proof can be extended to two dimensional quantum system through generalized measurement represented by positive operator-valued measured (POVM). In POVMs the number of available outcomes of a measurement may be higher than the dimensionality of the Hilbert space and N-outcome generalized measurement is represented by N-element POVM which consists of N positive semidefinite operators {}dE that sum to identity. Each pair of elements is not mutually orthogonal if the number of outcome of measurements is bigger than the dimensionality. In terms of POVM, Kochen-Specker theorem asserts that and could not be satisfied for . We developed a general model that enables us to generate different sizes of the POVM for the proof of the Kochen-Specker theorem. We show that the current simplest Nakamura model is in fact a special case of our model. W also provide another model which is as simple as the Nakamura’s but consists of different sets of POVM.