QuantumLogic 2009-01-29 16:41:52 2009-01-29 17:45:17 Topic theory of Hilbert lattices % this is the default PlanetPhysics preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amsmath, amssymb, amsfonts, amsthm, amscd, latexsym, enumerate} \usepackage{xypic, xspace} \usepackage[mathscr]{eucal} \usepackage[dvips]{graphicx} \usepackage[curve]{xy} % define commands here \theoremstyle{plain} \newtheorem{lemma}{Lemma}[section] \newtheorem{proposition}{Proposition}[section] \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}{Corollary}[section] \theoremstyle{definition} \newtheorem{definition}{Definition}[section] \newtheorem{example}{Example}[section] %\theoremstyle{remark} \newtheorem{remark}{Remark}[section] \newtheorem*{notation}{Notation} \newtheorem*{claim}{Claim} \renewcommand{\thefootnote}{\ensuremath{\fnsymbol{footnote}}} 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\newcommand{\rat}{{\rightarrowtail}} \newcommand{\ovset}[1]{\overset {#1}{\ra}} \newcommand{\ovsetl}[1]{\overset {#1}{\lra}} \newcommand{\hr}{{\hookrightarrow}} \newcommand{\<}{{\langle}} %\newcommand{\>}{{\rangle}} %\usepackage{geometry, amsmath,amssymb,latexsym,enumerate} %\usepackage{xypic} \def\baselinestretch{1.1} \hyphenation{prod-ucts} %\grpeometry{textwidth= 16 cm, textheight=21 cm} \newcommand{\sqdiagram}[9]{$$ \diagram #1 \rto^{#2} \dto_{#4}& #3 \dto^{#5} \\ #6 \rto_{#7} & #8 \enddiagram \eqno{\mbox{#9}}$$ } \def\C{C^{\ast}} \newcommand{\labto}[1]{\stackrel{#1}{\longrightarrow}} %\newenvironment{proof}{\noindent {\bf Proof} }{ \hfill $\Box$ %{\mbox{}} \newcommand{\quadr}[4] {\begin{pmatrix} & #1& \\[-1.1ex] #2 & & #3\\[-1.1ex]& #4& \end{pmatrix}} \def\D{\mathsf{D}} This is a contributed topic on \PMlinkexternal{quantum logic using tools available on the internet.}{http://us.metamath.org/qlegif/mmql.html} \section{Quantum Logic description} There are several approaches to quantum logic, and it should be therefore more appropriately called `Quantum Logics'. The following is a short list of such approaches to quantum logics. \begin{itemize} \item A standard approach to quantum logics is to add axioms so that it can be treated as a \emph{theory of Hilbert lattices}. Both Hilbert lattices and Hilbert space are involved in the foundation of quantum mechanics, and they are considered to be `dual' to each other. ``Just as Fourier transforms have led to greater insight into the nature of electrical signals, it may be possible that (via Solèr's theorem) quantum logic and Hilbert lattices will lead to new results in quantum mechanics.'' The axioms of this standard version of quantum logic (QL) can be specified as three distinct groups of axioms: \begin{enumerate} \item the ortholattice axioms: ax-a1 to ax-a5, and ax-r1, ax-r2, ax-r4, ax-r5 ; for example: ax-a1 is: $$a= N(Na)= NNa $$, where N stands for the logical negation; ax-a2 and ax-a3 are respectively the commutativity and associativity axioms; the ax-r1 ro ax-r5 axioms are implication axioms, such as: $$[a =b] \Rightarrow [b=a] $$ for ax-r1. \item the orthomodular law, ax-r3, that holds for those ortholattices that are also orthomodular lattices: $$1= [a \equiv b] \Rightarrow [a=b];$$ (interestingly, without ax-r3, the quantum logic becomes decidable), and \item stronger axioms than 1. and 2. for orthomodular lattices that are also Hilbert lattices. \end{enumerate} \begin{remark} The set of closed subspaces of a Hilbert space, $\mathcal{C}_H$ determines a special case of an orthomodular lattice $\left\langle{\mathcal{A},\cup , N}\right\rangle$. \end{remark} \item An interesting system for further studies is that in which the orthomodular lattice axiom or `orthomodular law', ax-r3, is replaced by a weaker axiom called the \emph{weakly orthomodular (WOM) law}; \item {\em Quantum propositional calculus}: quantum logic can be expressed and studied as a propositional calculus but involving the axioms or rules of quantum logics instead of those of Boolean logic. Quantum propositional calculus (QPC) is based on the algebra(s) of orthomodular lattices, similarly to the foundation of classical propositional calculus (CPC) on Boolean algebras. However, one notes that classical propositional calculus can also be modeled by a non-Boolean lattice, such as a centered $LM_n$-logic algebra. Another remarkable example is that of the logic lattice $O6$ ([0, a, b, Na, Nb,1]) which is a {\em non-distributive} model for classical propositional calculus. \item A second approach preferred by logicians is to define quantum logics via many-valued (MV) logic algebras such as the \L{}ukasiewicz-Moisil n-valued logic algebras. \end{itemize} \begin{thebibliography}{99} \bibitem{Kalmbach} Gudrun Kalmbach, ``{\em Orthomodular Lattices}'', Academic Press, London (1983). \bibitem{Beran} Ladislav Beran, Orthomodular Lattices; Algebraic Approach, D. Reidel, Dordrecht (1985). \bibitem{Pavicic1987} M. Pavici\'c, ``Minimal Quantum Logic with Merged Implications,'' Int. J. of Theor. Phys. 26, 845-852 (1987). \bibitem{PavMeg1998b} M. Pavici\'c and N. Megill, ``Quantum and Classical Implicational Algebras with Primitive Implication,'' \PMlinkexternal{Int. J. of Theor. Phys. 37, 2091-2098 (1998),}{ftp://m3k.grad.hr/pavicic/quantum-logic/1998-int-j-theor-phys-2.ps.gz}. \bibitem{PavMeg1999} M. Pavici\'c and N. Megill, ``Non-Orthomodular Models for Both Standard Quantum Logic and Standard Classical Logic: Repercussions for Quantum Computers,'' \PMlinkexternal{Helv. Phys. Acta 72, 189-210 (1999)}{http://xxx.lanl.gov/abs/quant-ph/9906101}. \bibitem{MegPav2000} N. Megill and M. Pavici\'c, ``Equations, States, and Lattices of Infinite-Dimensional Hilbert Space,'' \PMlinkexternal{Int. J. Theor. Phys. 39, 2337–2379 (2000),} {http://xxx.lanl.gov/abs/quant-ph/0009038}. \bibitem{McKMegPav} B. McKay, N. Megill, and M. Pavici\'c, ``Algorithms for Greechie Diagrams,'' \PMlinkexternal{Int. J. Theor. Phys. 39, 2393-2417(2000)}{http://xxx.lanl.gov/abs/quant-ph/0009039}. \bibitem{MegPav2001} N. Megill and M. Pavici\'c, ``Orthomodular Lattices and a Quantum Algebra,'' \PMlinkexternal{Int. J. Theor. Phys. 40, 1387-1410 (2001)}{http://xxx.lanl.gov/abs/quant-ph/0103135}. \end{thebibliography}