Orthomodular Lattices and a Quantum Algebra

Abstract

We show that one can formulate an algebra with lattice ordering so as to contain one quantum and five classical operations as opposed to the standard formulation of the Hilbert space subspace algebra. The standard orthomodular lattice is embeddable into the algebra. To obtain this result we devised algorithms and computer programs for obtaining expressions of all quantum and classical operations within an orthomodular lattice in terms of each other, many of which are presented in the paper. For quantum disjunction and conjunction we prove their associativity in an orthomodular lattice for any triple in which one of the elements commutes with the other two and their distributivity for any triple in which a particular one of the elements commutes with the other two. We also prove that the distributivity of symmetric identity holds in Hilbert space, although it remains an open problem whether it holds in all orthomodular lattices, as it does not fail in any of over 50 million Greechie diagrams we tested.