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Czechoslovak Mathematical Journal

State space properties of finite logics

Mirko Navara

Published Jan 1, 1987• 15 citations
Publisher

Abstract

We show that certain hypergraphs can be associated with quantum logics in a state-preserving manner.This result -a generalization of the Greechie cor respondence (see [2], [3]) -enables us to construct logics with special state space properties (see the applications in § 3).
DOI
10.21136/cmj.1987.102148

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Let LLL be a quantum logic and let S(L)S(L)S(L) denote the set of all states on LLL. (By a state we mean a nonnegative bounded σ\sigmaσ-additive measure, not necessarily normalized.) We ask whether every logic whose state space is a lattice has to be Boolean. We prove that this is so for finite logics and "projection logics." On the other hand, we show that there exist even concrete non-Boolean logics with a lattice state space (in fact, we prove that every countable concrete logic can be enlarged to a logic with a lattice state space). In the appendix we shortly consider the lattice properties of the set of observables and correct the paper [10].
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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