NON-ORTHOMODULAR MODELS FOR BOTH STANDARD QUANTUM LOGIC AND STANDARD CLASSICAL LOGIC: REPERCUSSIONS FOR QUANTUM COMPUTERS

Mladen Pavicic¹ and Norman D. Megill²

¹ Department of Mathematics, University of Zagreb, GF, Kaciceva 26, POB-217, HR-10001 Zagreb, Croatia; mpavicic@faust.irb.hr; http://m3k.grad.hr/pavicic

² nm@alum.mit.edu

Abstract. It is shown that propositional calculuses of both quantum and classical logics are non-categorical. We find that quantum logic is in addition to an orthomodular lattice also modeled by a weakly orthomodular lattice and that classical logic is in addition to a Boolean algebra also modeled by a weakly distributive lattice. Both new models turn out to be non-orthomodular. We prove the soundness and completeness of the calculuses for the models. We also prove that all the operations in an orthomodular lattice are five-fold defined. In the end we discuss possible repercussions of our results to quantum computations and quantum computers.

PACS numbers: 03.65.Bz, 02.10.By, 02.10.Gd

Keywords: quantum logic, orthomodular lattices, weakly orthomodular lattices, classical logic, Boolean algebra, weakly distributive lattices, non-categoricity, quantum computation.