[Paper Review] Orthomodular Lattices and a Quantum Algebra

Norman D. Megill, Mladen Pavičić|ArXiv.org|Mar 24, 2001

Advanced Algebra and Logic17 references4 citations

TL;DR

This paper introduces a novel quantum algebra (QA) that unifies five quantum operations with one classical operation within an orthomodular lattice, demonstrating that classical logic operations can be derived from quantum ones. Using computer-assisted algorithms, the authors prove associativity and distributivity under commutativity conditions and show that key quantum identities hold in Hilbert space and extensive Greechie diagrams, leaving full distributivity in all orthomodular lattices as an open problem.

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.

Motivation & Objective

• To reformulate orthomodular lattices using a single quantum operation as a foundational algebraic structure.
• To demonstrate that classical logical operations can be derived from quantum operations in orthomodular lattices.
• To prove associativity and distributivity of quantum operations under commutativity conditions.
• To investigate the validity of quantum distributive identities in Hilbert space and over 50 million Greechie diagrams.
• To explore whether quantum identities hold in weaker lattices like weakly orthomodular lattices (WOML).

Proposed method

• Developed computer algorithms to reduce two-variable expressions and derive equivalences between all quantum and classical operations in orthomodular lattices.
• Used a substitution rule to define a new quantum algebra (QA) based on a single quantum disjunction (Cup), replacing standard axioms.
• Proved that classical disjunction can be expressed in a single equation valid for all five quantum disjunctions (i=0,…,5).
• Established conditional associativity for quantum disjunction and conjunction when one element commutes with the other two.
• Verified distributive identities in Godowski lattices and tested over 50 million Greechie diagrams to assess generality.
• Provided algorithms and proofs for all derived identities, with key results embedded in lemmas and theorems.

Experimental results

Research questions

• RQ1 Can classical logical operations be uniquely derived from quantum operations within orthomodular lattices?
• RQ2 Does the distributive law for quantum identity hold in all orthomodular lattices, or only in specific classes like Hilbert space?
• RQ3 Is the conditional associativity of quantum disjunction and conjunction valid when one element commutes with the other two?
• RQ4 Can a quantum algebra be defined using a single quantum operation, with classical operations as derived entities?
• RQ5 Do certain quantum identities that hold in Hilbert space and in all tested Greechie diagrams also hold in weaker lattices like WOML?

Key findings

• Classical disjunction can be expressed in a single equation valid for all five quantum disjunctions (i=0,…,5), demonstrating a deep unification.
• Associativity of quantum disjunction and conjunction holds in any orthomodular lattice if one element commutes with the other two.
• Distributivity of the quantum identity holds in Godowski lattices and thus in Hilbert space, though it remains unproven for all orthomodular lattices.
• Over 50 million Greechie diagrams were tested, and no counterexample was found for the distributive law or the two identities in Eqs. (7.5) and (7.7).
• The quantum algebra QA is defined solely by a substitution rule, with standard axioms (A1–A7) being consequences rather than foundational.
• The paper identifies open problems, including finite axiomatization of QA and the validity of identities in weakly orthomodular lattices (WOML).