Non-Orthomodular Models for Both Standard Quantum Logic and Standard Classical Logic: Repercussions for Quantum Computers

It is shown that propositional calculuses of both quantum and classical logics are non- categorical. We nd 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 v e-fold dened. In the end we discuss possible repercussions of our results to quantum computations and quantum computers.

[1]  C. Ross.  Found , 1869, The Dental register.

[2]  Emil L. Post.  Introduction to a General Theory of Elementary Propositions , 1921 .

[3]  D. Hilbert,et al.  Principles of Mathematical Logic , 1950 .

[4]  S. C. Kleene,et al.  Introduction to Metamathematics , 1952 .

[5]  Ernst-Walther Stachow.  Completeness of quantum logic , 1976, J. Philos. Log..

[6]  H. Dishkant,et al.  Logic of Quantum Mechanics , 1976 .

[7]  Logikkalküle der Quantenphysik , 1977 .

[8]  Gary M. Hardegree.  The Conditional in Abstract and Concrete Quantum Logic , 1979 .

[9]  Ladislav Beran,et al.  Orthomodular Lattices: Algebraic Approach , 1985 .

[11]  M. Pavicic.  Unified quantum logic , 1989 .

[12]  Hugues Leblanc,et al.  Alternatives to Standard First-Order Semantics , 1989 .

[13]  Shirley Dex,et al.  JR 旅客販売総合システム（マルス）における運用及び管理について , 1991 .

[14]  M. Pavicic,et al.  Nonordered quantum logic and its YES-NO representation , 1993 .

[15]  S. Holland,et al.  Orthomodularity in infinite dimensions; a theorem of M. Solèr , 1995 .

[16]  Barenco,et al.  Elementary gates for quantum computation. , 1995, Physical review. A, Atomic, molecular, and optical physics.

[17]  Renatus Ziegler,et al.  On the Foundations of Set Theory , 1996 .

[18]  Quantum and Classical Implication Algebras with Primitive Implications , 1998 .

[19]  Identity Rule for Classical and Quantum Theories , 1998 .

[20]  Norman D. Megill,et al.  BINARY ORTHOLOGIC WITH MODUS PONENS IS EITHER ORTHOMODULAR OR DISTRIBUTIVE , 1998 .

[21]  Karl Svozil,et al.  Quantum Logic , 1998, Discrete mathematics and theoretical computer science.