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Isomorph-Free Exhaustive Generation of Greechie Diagrams and Automated Checking of Their Passage by Orthomodular Lattice Equations

Brendan D. McKay, Norman D. Megill, Mladen Paviÿcic

Published Sep 9, 2000• 4 citations
OpenAlex

Abstract

Abstract. We give a new algorithm for generating Greechie diagrams with arbitrary chosen number of atoms or blocks (with 2,3,4,... atoms) and provide a computer program for generating the diagrams. The results show that the previous algorithm does not produce every diagram and that it is at least 10 5 times slower. We also provide an algorithm and programs for checking of Greechie diagram passage by equations defining varieties of orthomodular lattices and give examples from Hilbert lattices. At the end we discuss some additional characteristics of Greechie diagrams. PACS numbers: 03.65.Bz, 02.10.By, 02.10.Gd

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The present article describes a method for checking the validity of implications or equivalences in the free orthomodular lattice on two generators and in the F(a, 6, ci,...,Cn), which is the free orthomodular lattice generated by the elements a, b, ci,... c", where the elements Ci, i = 1,..., n are central in it.The structure of the previous lattices is described in [3] and [1].The method presented is based on comparing the elements that are assigned to each expression on both sides of an implication or an equivalence.It gives a necessary condition for the implication or equivalence of arbitrary positive statements (a combination of identities and logical connectives AND and OR) to hold.When the conclusion part is an identity or a conjunction of identities, these conditions become also sufficient.

Demonstratio Mathematica•Oct 1, 2005•8 citations
Abstract. We give a new algorithm for generating Greechie diagrams with arbitrary chosen number of atoms or blocks (with 2,3,4,... atoms) and provide a computer program for generating the diagrams. The results show that the previous algorithm does not produce every diagram and that it is at least 10 5 times slower. We also provide an algorithm and programs for checking of Greechie diagram passage by equations defining varieties of orthomodular lattices and give examples from Hilbert lattices. At the end we discuss some additional characteristics of Greechie diagrams. PACS numbers: 03.65.Bz, 02.10.By, 02.10.Gd
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.
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.
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.
Using a graph approach to quantum systems, we show that descriptions of 3-dim Kochen–Specker (KS) setups as well as descriptions of 3-dim spin systems by means of Greechie diagrams (a kind of lattice) that we find in the literature are wrong. Correct lattices generated by McKay-Megill-Pavicic (MMP) hypergraphs and Hilbert subspace equations are given. To enable future exhaustive generation of 3-dim KS setups by means of our recently found stripping technique, bipartite graph generation is used to provide us with lattices with equal numbers of elements and blocks (orthogonal triples of elements)—up to 41 of them. We obtain several new results on such lattices and hypergraphs, in particular, on properties such as superposition and orthoraguesian equations.
The present article describes a method for checking the validity of implications or equivalences in the free orthomodular lattice on two generators and in the F(a, 6, ci,...,Cn), which is the free orthomodular lattice generated by the elements a, b, ci,... c", where the elements Ci, i = 1,..., n are central in it.The structure of the previous lattices is described in [3] and [1].The method presented is based on comparing the elements that are assigned to each expression on both sides of an implication or an equivalence.It gives a necessary condition for the implication or equivalence of arbitrary positive statements (a combination of identities and logical connectives AND and OR) to hold.When the conclusion part is an identity or a conjunction of identities, these conditions become also sufficient.