Greechie Diagrams of Small Quantum Logics with Small State Spaces

Mirko Navara, Pavel Pták

Jiří Binder

Karl Svozil, Josef Tkadlec

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.

Thurlow A. Cook

Wawrzyniec Guz

Mirko Navara

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).