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Kochen-Specker Sets and Generalized Orthoarguesian Equations

Norman D. Megill

Published Aug 16, 2016• 5 citations
OpenAlex

Abstract

We prove that the 7oa class (equational variety) of generalized orthoarguesian lattices is properly included in all noa classes for n < 7. This result strengthens the conjecture that any generalized orthoarguesian equation is strictly stronger than those of lower orders. The result emerged from our recent analysis of whether three-dimensional Kochen–Specker sets can be represented by Greechie lattices, which are a kind of orthomodular lattice.

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There are five known classes of lattice equations that hold in every infinite dimensional Hilbert space underlying quantum systems: generalised orthoarguesian, Mayet's E_A, Godowski, Mayet-Godowski, and Mayet's E equations. We obtain a result which opens a possibility that the first two classes coincide. We devise new algorithms to generate Mayet-Godowski equations that allow us to prove that the fourth class properly includes the third. An open problem related to the last class is answered. Finally, we show some new results on the Godowski lattices characterising the third class of equations.

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Greechie diagrams, nonexistence of measures in quantum logics, and Kochen–Specker-type constructions

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We use Greechie diagrams to construct finite orthomodular lattices ‘‘realizable’’ in the orthomodular lattice of subspaces in a three-dimensional Hilbert space such that the set of two-valued states is not ‘‘large’’ (i.e., full, separating, unital, nonempty, resp.). We discuss the number of elements of such orthomodular lattices, of their sets of (ortho)generators and of their subsets that do not admit a ‘‘large’’ set of two-valued states. We show connections with other results of this type.

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Algebraic partial Boolean algebras

D. A. Smith

Partial Boolean algebras, first studied by Kochen and Specker in the 1960s, provide the structure for Bell–Kochen–Specker theorems which deny the existence of non-contextual hidden variable theories. In this paper, we study partial Boolean algebras which are ‘algebraic’ in the sense that their elements have coordinates in an algebraic number field. Several of these algebras have been discussed recently in a debate on the validity of Bell–Kochen–Specker theorems in the context of finite precision measurements. The main result of this paper is that every algebraic finitely-generated partial Boolean algebra B ( T ) is finite when the underlying space ℋ is three-dimensional, answering a question of Kochen and showing that Conway and Kochen's infinite algebraic partial Boolean algebra has minimum dimension. This result contrasts the existence of an infinite (non-algebraic) B ( T ) generated by eight elements in an abstract orthomodular lattice of height 3. We then initiate a study of higher-dimensional algebraic partial Boolean algebras. First, we describe a restriction on the determinants of the elements of B ( T ) that are generated by a given set T . We then show that when the generating set T consists of the rays spanning the minimal vectors in a real irreducible root lattice, B ( T ) is infinite just if that root lattice has an A 5 sublattice. Finally, we characterize the rays of B ( T ) when T consists of the rays spanning the minimal vectors of the root lattice E 8 .

Journal of Physics A Mathematical and General•Mar 19, 2003•5 citations
We prove that the 7oa class (equational variety) of generalized orthoarguesian lattices is properly included in all noa classes for n < 7. This result strengthens the conjecture that any generalized orthoarguesian equation is strictly stronger than those of lower orders. The result emerged from our recent analysis of whether three-dimensional Kochen–Specker sets can be represented by Greechie lattices, which are a kind of orthomodular lattice.
We provide several new results on quantum state space, on lattice of subspaces of an infinite dimensional Hilbert space, and on infinite dimensional Hilbert space equations as well as on connections between them. In particular we obtain an n-variable generalized orthoarguesian equation which holds in any infinite dimensional Hilbert space. Then we strengthen Godowski's result by showing that in an ortholattice on which strong states are defined Godowski's equations as well as the orthomodularity hold. We also prove that all 6- and 4-variable orthoarguesian equations presented in the literature can be reduced to new 4- and 3-variable ones, respectively and that Mayet's examples follow from Godowski's equations. To make a breakthrough in testing these massive equations we designed several novel algorithms for generating Greechie diagrams with an arbitrary number of blocks and atoms (currently testing with up to 50) and for automated checking of equations on them. A way of obtaining complex infinite dimensional Hilbert space from the Hilbert lattice equipped with several additional conditions and without invoking the notion of state is presented. Possible repercussions of the results to quantum computing problems are discussed.
Abstract. We provide several new results on quantum state space, on lattice of subspaces of an infinite dimensional Hilbert space, and on infinite dimensional Hilbert space equations as well as on connections between them. In particular we obtain an n-variable generalized orthoarguesian equation which holds in any infinite dimensional Hilbert space. Then we strengthen Godowski’s result by showing that in an ortholattice on which strong states are defined Godowski’s equations as well as the orthomodularity hold. We also prove that all 6- and 4-variable orthoarguesian equations presented in the literature can be reduced to new 4- and 3-variable ones, respectively and that Mayet’s examples follow from Godowski’s equations. To make a breakthrough in testing these massive equations we designed several novel algorithms for generating Greechie diagrams with an arbitrary number of blocks and atoms (currently testing with up to 50) and for automated checking of equations on them. A way of obtaining complex infinite dimensional Hilbert space from the Hilbert lattice equipped with several additional conditions and without invoking the notion of state is presented. Possible repercussions of the results to quantum computing problems are discussed.
There are five known classes of lattice equations that hold in every infinite dimensional Hilbert space underlying quantum systems: generalised orthoarguesian, Mayet's E_A, Godowski, Mayet-Godowski, and Mayet's E equations. We obtain a result which opens a possibility that the first two classes coincide. We devise new algorithms to generate Mayet-Godowski equations that allow us to prove that the fourth class properly includes the third. An open problem related to the last class is answered. Finally, we show some new results on the Godowski lattices characterising the third class of equations.
We use Greechie diagrams to construct finite orthomodular lattices ‘‘realizable’’ in the orthomodular lattice of subspaces in a three-dimensional Hilbert space such that the set of two-valued states is not ‘‘large’’ (i.e., full, separating, unital, nonempty, resp.). We discuss the number of elements of such orthomodular lattices, of their sets of (ortho)generators and of their subsets that do not admit a ‘‘large’’ set of two-valued states. We show connections with other results of this type.
Partial Boolean algebras, first studied by Kochen and Specker in the 1960s, provide the structure for Bell–Kochen–Specker theorems which deny the existence of non-contextual hidden variable theories. In this paper, we study partial Boolean algebras which are ‘algebraic’ in the sense that their elements have coordinates in an algebraic number field. Several of these algebras have been discussed recently in a debate on the validity of Bell–Kochen–Specker theorems in the context of finite precision measurements. The main result of this paper is that every algebraic finitely-generated partial Boolean algebra B ( T ) is finite when the underlying space ℋ is three-dimensional, answering a question of Kochen and showing that Conway and Kochen's infinite algebraic partial Boolean algebra has minimum dimension. This result contrasts the existence of an infinite (non-algebraic) B ( T ) generated by eight elements in an abstract orthomodular lattice of height 3. We then initiate a study of higher-dimensional algebraic partial Boolean algebras. First, we describe a restriction on the determinants of the elements of B ( T ) that are generated by a given set T . We then show that when the generating set T consists of the rays spanning the minimal vectors in a real irreducible root lattice, B ( T ) is infinite just if that root lattice has an A 5 sublattice. Finally, we characterize the rays of B ( T ) when T consists of the rays spanning the minimal vectors of the root lattice E 8 .