\documentclass[rmp,amsfonts,showpacs,showkeys]{revtex4} %\documentclass[pra,showpacs,showkeys,amsfonts]{revtex4} \usepackage[T1]{fontenc} \usepackage{graphicx} %\documentstyle[amsfonts]{article} \RequirePackage{times} %\RequirePackage{courier} \RequirePackage{mathptm} %\renewcommand{\baselinestretch}{1.3} \def\frak{\bf} \def\Bbb{\bf} \def\sp{\textrm{Sp}} \def\hh{{\frak H}} \def\ll{{\frak C (\frak H)}} \def\mm{{\frak M}} \def\ppp{{\frak P} (\frak H)} \begin{document} %\sloppy \title{Quantum logic. A brief outline} \author{Karl Svozil} \email{svozil@tuwien.ac.at} \homepage{http://tph.tuwien.ac.at/~svozil} \affiliation{Institut f\"ur Theoretische Physik, University of Technology Vienna, Wiedner Hauptstra\ss e 8-10/136, A-1040 Vienna, Austria} \begin{abstract} Quantum logic has been introduced by Birkhoff and von Neumann as an attempt to base the logical primitives, the propositions and the relations and operations among them, on quantum theoretical entities, and thus on the related empirical evidence of the quantum world. We give a brief outline of quantum logic, and some of its algebraic properties, such as nondistributivity, whereby emphasis is given to concrete experimental setups related to quantum logical entities. A probability theory based on quantum logic is fundamentally and sometimes even spectacularly different from probabilities based on classical Boolean logic. We give a brief outline of its nonclassical aspects; in particular violations of Boole-Bell type consistency constraints on joint probabilities, as well as the Kochen-Specker theorem, demonstrating in a constructive, finite way the scarcity and even nonexistence of two-valued states interpretable as classical truth assignments. {A more complete introduction of the author can be found in the book {\it Quantum Logic (Springer, 1998)}} \end{abstract} \pacs{03.67.Hk,03.65.Ud,03.65.Ta,03.67.Mn} \keywords{Quantum logic, quantum information, quantum probabilities} \maketitle % \tableofcontents \section{Basic ideas} Quantum logic has been introduced by Garrett Birkhoff and John von Neumann in the thirties \cite{birkhoff-36}. They organized it {\em top-down}: The starting point is von Neumann's Hilbert space formalism of quantum mechanics. In a second step, certain entities of Hilbert spaces are identified with propositions, partial order relations and lattice operations --- Birkhoff's field of expertise. These relations and operations can then be associated with the logical implication relation and operations such as {\it and}, {\it or}, and {\it not}. Thereby, a ``nonclassical,'' nonboolean logical structure is induced which originates in theoretical physics. If theoretical physics is taken as a faithful representation of our experience, such an ``operational'' \cite{bridgman27,bridgman,bridgman52} logic derives its justification by the phenomena themselves. In this sense, one of the main ideas behind quantum logic is the quasi-inductive construction of the logical and algebraic order of events from empirical facts. This is very different from the ``classical'' logical approach, which is also {\em top-down}: There, the system of symbols, the axioms, as well as the rules of inference are mentally constructed, abstract objects of our thought. Insofar as our thoughts can pretend to exist independent from the physical Universe, such ``classical'' logical systems can be conceived as totally independent from the world of the phenomena. In the following we shall shortly review quantum logic. More detailed introduction can be found in the books of Mackey \cite{ma-57}, Jauch \cite{jauch}, Varadarajan \cite{varadarajanI,varadarajanII}, Piron \cite{piron-76}, Marlow \cite{marlow}, Gudder \cite{gudder:79,gudder}, Maczy\'nski \cite{maczy}, Beltrametti and Cassinelli \cite{bell-cas}, Kalmbach \cite{kalmbach-83,kalmbach-86}, Cohen \cite{cohen}, Pt{\'{a}}k and Pulmannov{\'{a}} \cite{pulmannova-91}, Giuntini \cite{giuntini-91}, and in a forthcoming book of the author \cite{svozil-ql}, among others. A bibliography on quantum logics and related structures has been compiled by Pavi{\v{c}}i{\'{c}} \cite{pavicic-92}. \begin{description} \item[$\bullet$] Any closed linear subspace of --- or, equivalently, any projection operator on --- a Hilbert space corresponds to an elementary proposition. The elementary {\it true}--{\it false} proposition can in English be spelled out explicitly as \begin{quote} ``The physical system has a property corresponding to the associated closed linear subspace.'' \end{quote} \item[$\bullet$] The logical {\it and} operation is identified with the set theoretical intersection of two propositions ``$\cap$''; i.e., with the intersection of two subspaces. It is denoted by the symbol ``$\wedge$''. So, for two propositions $p$ and $q$ and their associated closed linear subspaces ${\frak M}_p$ and ${\frak M}_q$, $$ {\frak M}_{p\wedge q} = \{x \mid x\in {\frak M}_p, \; x\in {\frak M}_q\} .$$ \item[$\bullet$] The logical {\it or} operation is identified with the closure of the linear span ``$\oplus$'' of the subspaces corresponding to the two propositions.\footnote{ Notice that a vector of Hilbert space may be an element of $ {\frak M}_{p} \oplus {\frak M}_{q} $ without being an element of either $ {\frak M}_{p} $ or ${\frak M}_{q} $, since $ {\frak M}_{p} \oplus {\frak M}_{q} $ includes all the vectors in $ {\frak M}_{p} \cup {\frak M}_{q} $, as well as all of their linear combinations (superpositions) and their limit vectors. } It is denoted by the symbol ``$\vee$''. So, for two propositions $p$ and $q$ and their associated closed linear subspaces ${\frak M}_p$ and ${\frak M}_q$, $$ {\frak M}_{p\vee q} = {\frak M}_{p} \oplus {\frak M}_{q} = \{x \mid x=\alpha x+\beta z,\; \alpha,\beta \in {\Bbb C},\; y\in {\frak M}_p, \; z\in {\frak M}_q\} .$$ The symbol $\oplus$ will used to indicate the closed linear subspace spanned by two vectors. That is, $$u\oplus v=\{ w\mid w=\alpha u+ \beta v,\; \alpha,\beta \in {\Bbb C} ,\; u,v \in {\frak H}\}.$$ More generally, the symbol $\oplus$ indicates the closed linear subspace spanned by two linear subspaces. That is, if $u,v\in {\frak C}({{\frak H}})$, where ${\frak C}({{\frak H}})$ stands for the set of all subspaces of the Hilbert space, then $$u\oplus v=\{ w\mid w=\alpha u+ \beta v,\; \alpha,\beta \in {\Bbb R} ,\; u,v \in {\frak C}({{\frak H}}) \}.$$ \item[$\bullet$] The logical {\it not}-operation, or the ``complement'', is identified with operation of taking the orthogonal subspace ``$\perp$''. It is denoted by the symbol ``~$'$~''. In particular, for a proposition $p$ and its associated closed linear subspace ${\frak M}_p$, $$ {\frak M}_{p'} = \{x \mid (x,y)=0,\; y\in {\frak M}_p \} .$$ \item[$\bullet$] The logical {\it implication} relation is identified with the set theoretical subset relation ``$\subset$''. It is denoted by the symbol ``$\rightarrow$''. So, for two propositions $p$ and $q$ and their associated closed linear subspaces ${\frak M}_p$ and ${\frak M}_q$, $$ {p\rightarrow q} \Longleftrightarrow {\frak M}_{p} \subset {\frak M}_{q}.$$ \item[$\bullet$] A trivial statement which is always {\it true} is denoted by $1$. It is represented by the entire Hilbert space $\frak H$. So, $${\frak M}_1=\frak H.$$ \item[$\bullet$] An absurd statement which is always {\it false} is denoted by $0$. It is represented by the zero vector $0$. So, $${\frak M}_0= 0.$$ \end{description} Let us verify some logical statements. \begin{description} \item[$(p')'=p$:] For closed orthogonal subspaces of ${\frak H}$, $ {\frak M}_{(p')'} = \{x\mid (x,y)=0, y\in \{z\mid (z,u)=0, u\in {\frak M}_{p} \} \} = {\frak M}_{p}. $ \item[$1'=0$:] $ {\frak M}_{1'} = \{x\mid (x,y)=0, y\in {\frak H}\} ={\frak M}_0= 0. $ \item[$0'=1$:] $ {\frak M}_{0'} = \{x\mid (x,y)=0, y=0\} ={\frak M}_1= {\frak H}. $ \item[$p\vee p'=1$:] $ {\frak M}_{p\vee p'}= {\frak M}_{p} \oplus {\frak M}_{p'} = \{x \mid x=\alpha y+\beta z,\; \alpha,\beta \in {\Bbb C},\; y\in {\frak M}_p, \; z\in {\frak M}_{p'}\} ={\frak M}_1 .$ \item[$p\wedge p'=0$:] $ {\frak M}_{p\wedge p'}= {\frak M}_{p} \cap {\frak M}_{p'} = \{x \mid x\in {\frak M}_p, \; x\in {\frak M}_{p'}\} ={\frak M}_0 .$ \end{description} Table \ref{tcompa} lists the identifications of relations of operations of various lattice types. \begin{table}[h] \begin{center} {\footnotesize \begin{tabular}{ccccc} \hline\hline generic lattice & order relation $\rightarrow $ & ``meet'' $\sqcap$ & ``join'' $\sqcup$ & ``complement'' $'$\\ \hline ``classical'' lattice & subset $\subset $ & intersection $\cap$ & union $\cup$ & complement\\ of subsets&&&&\\ of a set&&&&\\ \hline propositional&implication&disjunction&conjunction&negation\\ calculus&$\rightarrow$&``${\it and}$'' $\wedge$&``${\it or}$'' $\vee$&``${\it not}$''$\neg$\\ \hline Hilbert & subspace& intersection of & closure of & orthogonal \\ lattice & relation& subspaces $\cap$& linear& subspace \\ & $\subset$ & & span $\oplus$ & $\perp$ \\ \hline lattice of& $E_1E_2=E_1$& $E_1E_2$& $E_1+E_2-E_1E_2$& orthogonal\\ commuting&&($\lim_{n\rightarrow \infty}(E_1E_2)^n$)&&projection\\ (noncommuting)\\ projection\\ operators\\ \hline\hline \end{tabular} } \caption{Comparison of the identifications of lattice relations and operations for the lattices of subsets of a set, for experimental propositional calculi, for Hilbert lattices, and for lattices of commuting projection operators. \label{tcompa}} \end{center} \end{table} Propositional structures are often represented by Hasse and Greechie diagrams. A Hasse diagram is a convenient representation of the logical implication, as well as of the {\it and} and {\it or} operations among propositions. Points ``~$\bullet$~'' represent propositions. Propositions which are implied by other ones are drawn higher than the other ones. Two propositions are connected by a line if one implies the other. A much more compact representation of the propositional calculus can be given in terms of its Greechie diagram. There, the points ``~$\circ$~'' represent the atoms. \label{greechie-diagram} \index{Greechie diagram} If they belong to the same Boolean algebra, they are connected by edges or smooth curves. We will later use ``almost'' Greechie diagrams, omitting points which belong to only one curve. This makes the diagrams a bit more comprehensive. \section{Complementarity} Let us call two propositions $p,q$ {\em comeasurable} or {\em compatible} if and only if there exist mutually orthogonal propositions $a,b,c$ such that $p=a\vee b$ and $q=a\vee c$ \cite[p. 118]{varadarajanI}. Intuitively, we may assume that two comeasurable propositions $c\vee a$ and $c\vee b$ consist of an ``identical'' part $a$, as well as of the orthogonal parts $b,c$ (which are also orthogonal to $a$). Clearly, orthogonality implies comeasurability, since if $p$ and $q$ are orthogonal, we may identify $a, b, c$ with $0,p,q$, respectively. If one is willing to give meaning to noncomeasurable blocks of observables and thus to counterfactuals, it is straightforward to proceed with the formalism. Consider a collection of blocks. Some of these blocks may have a common nontrivial observable. The complete logic with respect to the collection of the blocks is obtained by the following construction. \index{pasting} \begin{description} \item[$\bullet$] The tautologies of all blocks are identified. \item[$\bullet$] The absurdities of all blocks are identified. \item[$\bullet$] Identical elements in different blocks are identified. \item[$\bullet$] The logical and algebraical structures of all blocks remain intact. \end{description} This construction is often referred to as {\em pasting} construction. If the blocks are only pasted together at the tautology and the absurdity, one calls the resulting logic a {\em horizontal sum}.\footnote{This definition of ``horizontal sum'' is equivalent to the coproduct of complemented lattices. A coproduct is like a least upper bound: each component maps onto it, and if each component maps into another complemented lattice, then the coproduct also does in a unique way.} \label{l-horsum} \index{horizontal sum} In a sense, the pasting construction allows one to obtain a {\em global} representation of different universes which are defined (and classical) {\em locally}. This local --- {\it versus} --- global theme will be discussed below. A pasting of two propositional structures $L_1$ and $L_2$ will be denoted by $$L_1\oplus L_2.$$ For further reading, see also Gudrun Kalmbach \cite{kalmbach-83,kalmbach-86}, Robert Piziak \cite{piz-88}, Pavel Pt{\'{a}}k and Sylvia Pulmannov{\'{a}} \cite{pulmannova-91}, and Mirko Navara and V. Rogalewicz \cite{nav:91}. \section{Spin one-half} As a first example, we shall paste together observables of the spin one-half systems. We have associated a propositional system $$L(x)= \{ 0, p_{-}, p_{+}, 1 \}, $$ corresponding to the outcomes of a measurement of the spin states along the $x$-axis (any other direction would have done as well). If the spin states would be measured along a different spatial direction, say $\overline{x}\neq x \; \textrm{mod} \pi$, an identical propositional system $$L(\overline x)= \{ \overline 0, \overline p_{-}, \overline p_{+}, \overline 1 \} $$ would have been the result. $L(x)$ and $L(\overline x)$ can be jointly represented by pasting them together. In particular, we identify their tautologies and absurdities; i.e., \begin{eqnarray*} 0 &=& \overline 0,\\ 1 &=& \overline 1. \end{eqnarray*} All the other propositions remain distinct. We then obtain a propositional structure $$L(x)\oplus L(\overline{x}) = MO_2$$ whose Hasse diagram is of the ``Chinese lantern'' form and is drawn in Figure \ref{f-hd-mo2}. The corresponding Greechie Diagram is drawn in Figure \ref{f-gd-mo2}. Here, the ``$O$'' stands for {\em orthocomplemented,} the term ``$M$'' stands for {\em modular} (cf. page \pageref{l-modular} below), and the subscript ``2'' stands for the pasting of two Boolean subalgebras $2^2$. \begin{figure}[htd] \begin{center} %TexCad Options %\grade{\off} %\emlines{\off} %\beziermacro{\off} %\reduce{\on} %\snapping{\off} %\quality{0.20} %\graddiff{0.01} %\snapasp{1} %\zoom{1.00} \unitlength 0.60mm \linethickness{0.4pt} \begin{picture}(125.00,60.73) \put(60.00,0.00){\circle*{2.11}} \put(30.00,30.00){\circle*{2.11}} \put(60.00,59.67){\circle*{2.11}} \put(90.00,30.00){\circle*{2.11}} \put(60.00,0.00){\line(-1,1){30.00}} \put(30.00,30.00){\line(1,1){30.00}} \put(60.00,60.00){\line(1,-1){30.00}} \put(90.00,30.00){\line(-1,-1){30.00}} \put(65.00,0.00){\makebox(0,0)[lc]{$0=1'$}} \put(65.00,60.00){\makebox(0,0)[lc]{$1=p_-\vee p_+=\overline p_-\vee\overline p_+$}} \put(30.00,25.00){\makebox(0,0)[cc]{$p_+$}} \put(90.00,25.00){\makebox(0,0)[cc]{$\overline p_-$}} \put(60.00,0.00){\line(-5,3){50.00}} \put(10.00,30.00){\line(5,3){50.00}} \put(60.00,60.00){\line(5,-3){50.00}} \put(110.00,30.00){\line(-5,-3){50.00}} \put(10.00,30.00){\circle*{2.11}} \put(10.00,25.00){\makebox(0,0)[cc]{$p_-$}} \put(110.00,30.00){\circle*{2.11}} \put(110.00,25.00){\makebox(0,0)[cc]{$\overline p_+$}} \put(25.00,50.00){\makebox(0,0)[cc]{$L(x)$}} \put(95.00,50.00){\makebox(0,0)[cc]{$L(\overline x)$}} \put(0.00,15.00){\dashbox{2.00}(45.00,30.00)[cc]{}} \put(80.00,15.00){\dashbox{2.00}(45.00,30.00)[cc]{}} \end{picture} \end{center} \caption{\label{f-hd-mo2} Hasse diagram of the ``Chinese lantern'' form obtained by the pasting of two spin one-half state propositional systems $L(x)$ and $L(\overline x)$ which are noncomeasurable. The resulting logical structure is a modular orthocomplemented lattice $L(x)\oplus L(\overline{x}) = MO_2$. \index{$MO_2$} The blocks (without $0,1$) are indicated by dashed boxes. They will be henceforth omitted. } \end{figure} \begin{figure}[htd] \begin{center} %TexCad Options %\grade{\off} %\emlines{\off} %\beziermacro{\off} %\reduce{\on} %\snapping{\off} %\quality{0.20} %\graddiff{0.01} %\snapasp{1} %\zoom{1.00} \unitlength 0.50mm \linethickness{0.4pt} \begin{picture}(131.06,15.00) \put(0.00,5.00){\circle{2.11}} \put(60.00,5.00){\circle{2.11}} \put(0.00,0.00){\makebox(0,0)[cc]{$p_-$}} \put(60.00,0.00){\makebox(0,0)[cc]{$p_+$}} \put(0.00,5.00){\line(1,0){60.00}} \put(70.00,5.00){\circle{2.11}} \put(130.00,5.00){\circle{2.11}} \put(70.00,0.00){\makebox(0,0)[cc]{$\overline p_-$}} \put(130.00,0.00){\makebox(0,0)[cc]{$\overline p_+$}} \put(70.00,5.00){\line(1,0){60.00}} \put(30.00,15.00){\makebox(0,0)[cc]{$L(x)$}} \put(100.00,15.00){\makebox(0,0)[cc]{$L(\overline x)$}} \end{picture} \end{center} \caption{\label{f-gd-mo2} Greechie diagram of two spin one-half state propositional systems $L(x)$ and $L(\overline x)$ which are noncomeasurable. } \end{figure} The propositional system obtained is {\em not} a classical Boolean algebra, since the distributive law is not satisfied. This can be easily seen by \index{distributivity} the following evaluation. Assume that the distributive law is satisfied. Then,\label{l-dist} \begin{eqnarray*} p_- \vee ({\overline p}_- \wedge {\overline p}_-') &=& (p_- \vee {\overline p}_-) \wedge (p_- \vee {\overline p}_-'),\\ p_- \vee 0 &=& 1 \wedge 1,\\ p_- &=& 1. \end{eqnarray*} This is incorrect. A similar calculation yields \begin{eqnarray*} p_- \wedge ({\overline p}_- \vee {\overline p}_-') &=& (p_- \wedge {\overline p}_-) \vee (p_- \wedge {\overline p}_-'),\\ p_- \wedge 1 &=& 0 \wedge 0,\\ p_- &=& 0, \end{eqnarray*} which again is incorrect. Notice that the expressions can be easily evaluated by using the Hasse diagram \ref{f-hd-mo2}. For any $a,b$, $a\vee b$ is just the least element which is connected by $a$ and $b$; $a\wedge b$ is just the highest element connected to $a$ and $b$. Intermediates which are not connected to both $a$ and $b$ do not count. That is, \begin{center} %TexCad Options %\grade{\off} %\emlines{\off} %\beziermacro{\off} %\reduce{\on} %\snapping{\off} %\quality{2.00} %\graddiff{0.01} %\snapasp{1} %\zoom{1.00} \unitlength 0.30mm \linethickness{0.4pt} \begin{picture}(105.00,20.95) \put(0.00,0.00){\line(1,1){20.00}} \put(20.00,20.00){\line(1,-1){20.00}} \put(60.00,20.00){\line(1,-1){20.00}} \put(80.00,0.00){\line(1,1){20.00}} \put(100.00,20.00){\circle*{1.89}} \put(60.00,20.00){\circle*{1.89}} \put(20.00,20.00){\circle*{1.89}} \put(80.00,0.00){\circle*{1.89}} \put(40.00,0.00){\circle*{1.89}} \put(0.00,0.00){\circle*{1.89}} \put(5.00,0.00){\makebox(0,0)[cc]{$a$}} \put(45.00,0.00){\makebox(0,0)[cc]{$b$}} \put(65.00,20.00){\makebox(0,0)[cc]{$a$}} \put(105.00,20.00){\makebox(0,0)[cc]{$b$}} \put(25.00,20.00){\makebox(0,0)[lc]{$a\vee b$}} \put(85.33,0.00){\makebox(0,0)[lc]{$a \wedge b$}} \end{picture} \end{center} $a\vee b$ is called a least upper bound of $a$ and $b$. \index{least upper bound} $a\wedge b$ is called a greatest lower bound of $a$ and $b$. \index{greatest lower bound} $MO_2$ is a specific example of an algebraic structure which is called a {\em lattice}. Any two elements of a lattice have a least upper and a greatest lower bound. Furthermore, \begin{eqnarray*} a\rightarrow b \textrm{ and } a \rightarrow c, &\textrm{ then }& a \rightarrow (b\wedge c);\\ b \rightarrow a \textrm{ and } c \rightarrow a, &\textrm{ then }& (b\vee c) \rightarrow a. \end{eqnarray*} It is an {\em ortholattice} or {\em orthocomplemented lattice}, since every element has a complement. It is modular, since for all $a\rightarrow c$, the modular law \index{modular} \label{l-modular} $$ (a\vee b)\wedge c= a\vee (b \wedge c) $$ is satisfied. For example, \begin{eqnarray*} (p_-\vee p_+)\wedge 1&=& p_-\vee (p_+ \wedge 1),\\ 1\wedge 1&=& p_-\vee p_+,\\ 1&=&1. \end{eqnarray*} One can proceed and consider a finite number $n$ of different directions of spin state measurements, corresponding to $n$ distinct orientations of a Stern-Gerlach apparatus. The resulting propositional structure is the horizontal sum $MO_n$ of $n$ classical Boolean algebras \index{$MO_n$} $L(x^i)$, where $x^i$ indicates the direction of a spin state measurement. That is, $$MO_n=\oplus_{i=1}^n L(x^i).$$ Figure \ref{f-hd-mon} and Figure \ref{f-gd-mon} show its Hasse and Greechie diagrams, respectively. Of course, it should be emphasized that only a {\em single} $L(x^i)$ can actually be operationalized. According to quantum mechanics, all the other $n-1$ unchecked quasi-classical worlds remain in permanent oblivion. \begin{figure}[htd] \begin{center} %TexCad Options %\grade{\off} %\emlines{\off} %\beziermacro{\off} %\reduce{\on} %\snapping{\off} %\quality{0.20} %\graddiff{0.01} %\snapasp{1} %\zoom{0.90} \unitlength 0.60mm \linethickness{0.4pt} \begin{picture}(171.67,126.67) \put(85.00,0.00){\line(1,4){10.00}} \put(95.00,40.00){\line(-1,4){9.92}} \put(85.08,79.67){\line(3,-4){29.75}} \put(114.83,40.00){\line(-3,-4){30.00}} \put(84.83,0.00){\line(5,4){50.17}} \put(135.00,40.13){\line(-5,4){50.00}} \put(85.00,0.00){\line(5,3){67.00}} \put(152.00,40.20){\line(-5,3){67.00}} \put(85.00,80.00){\circle*{3.33}} \put(95.00,40.00){\circle*{3.33}} \put(115.00,40.00){\circle*{3.33}} \put(135.00,40.00){\circle*{3.33}} \put(152.33,40.00){\circle*{3.33}} \put(85.00,-0.33){\circle*{3.33}} \put(93.33,80.00){\makebox(0,0)[lc]{$1$}} \put(93.33,0.00){\makebox(0,0)[lc]{$0$}} \put(17.67,35.00){\makebox(0,0)[cc]{$p_-^1$}} \put(34.67,35.00){\makebox(0,0)[cc]{$p_+^1$}} \put(55.00,35.00){\makebox(0,0)[cc]{$p_-^2$}} \put(74.00,35.00){\makebox(0,0)[cc]{$p_+^2$}} \put(97.00,35.00){\makebox(0,0)[cc]{$p_-^3$}} \put(115.67,35.00){\makebox(0,0)[cc]{$p_+^3$}} \put(135.00,35.00){\makebox(0,0)[cc]{$p_-^n$}} \put(152.67,35.00){\makebox(0,0)[cc]{$p_+^n$}} \put(14.67,95.00){\circle*{3.33}} \put(-0.33,110.00){\circle*{3.33}} \put(29.67,110.00){\circle*{3.33}} \put(14.67,125.00){\circle*{3.33}} \put(-0.33,110.00){\line(1,1){15.00}} \put(14.67,125.00){\line(1,-1){15.00}} \put(29.67,110.00){\line(-1,-1){15.00}} \put(14.67,95.00){\line(-1,1){15.00}} \put(21.00,95.00){\makebox(0,0)[cc]{$0^1 $}} \put(21.00,125.00){\makebox(0,0)[cc]{$1^1$}} \put(-0.33,104.67){\makebox(0,0)[cc]{$p_-^1$}} \put(29.67,105.00){\makebox(0,0)[cc]{$p_+^1$}} \put(35.50,110.00){\makebox(0,0)[cc]{$\oplus$}} \put(55.92,95.00){\circle*{3.33}} \put(98.83,95.00){\circle*{3.33}} \put(155.00,95.00){\circle*{3.33}} \put(40.92,110.00){\circle*{3.33}} \put(83.83,110.00){\circle*{3.33}} \put(140.00,110.00){\circle*{3.33}} \put(70.92,110.00){\circle*{3.33}} \put(113.83,110.00){\circle*{3.33}} \put(170.00,110.00){\circle*{3.33}} \put(55.92,125.00){\circle*{3.33}} \put(98.83,125.00){\circle*{3.33}} \put(155.00,125.00){\circle*{3.33}} \put(40.92,110.00){\line(1,1){15.00}} \put(83.83,110.00){\line(1,1){15.00}} \put(140.00,110.00){\line(1,1){15.00}} \put(55.92,125.00){\line(1,-1){15.00}} \put(98.83,125.00){\line(1,-1){15.00}} \put(155.00,125.00){\line(1,-1){15.00}} \put(70.92,110.00){\line(-1,-1){15.00}} \put(113.83,110.00){\line(-1,-1){15.00}} \put(170.00,110.00){\line(-1,-1){15.00}} \put(55.92,95.00){\line(-1,1){15.00}} \put(98.83,95.00){\line(-1,1){15.00}} \put(155.00,95.00){\line(-1,1){15.00}} \put(62.25,95.00){\makebox(0,0)[cc]{$0^2 $}} \put(105.16,95.00){\makebox(0,0)[cc]{$0^3$}} \put(161.33,95.00){\makebox(0,0)[cc]{$0^n$}} \put(62.25,125.00){\makebox(0,0)[cc]{$1^2$}} \put(105.16,125.00){\makebox(0,0)[cc]{$1^3$}} \put(161.33,125.00){\makebox(0,0)[cc]{$1^n$}} \put(40.92,104.67){\makebox(0,0)[cc]{$p_-^2$}} \put(83.83,104.67){\makebox(0,0)[cc]{$p_-^3$}} \put(140.00,104.67){\makebox(0,0)[cc]{$p_-^n$}} \put(70.92,105.00){\makebox(0,0)[cc]{$p_+^2$}} \put(113.83,105.00){\makebox(0,0)[cc]{$p_+^3$}} \put(170.00,105.00){\makebox(0,0)[cc]{$p_+^n$}} \put(77.58,110.00){\makebox(0,0)[cc]{$\oplus$}} \put(119.67,110.00){\makebox(0,0)[cc]{$\oplus$}} \put(134.67,110.00){\makebox(0,0)[cc]{$\oplus$}} \put(121.33,40.00){\circle*{0.67}} \put(123.00,40.00){\circle*{0.67}} \put(124.67,40.00){\circle*{0.67}} \put(125.50,110.00){\circle*{0.67}} \put(127.17,110.00){\circle*{0.67}} \put(128.84,110.00){\circle*{0.67}} \put(5.00,40.00){\makebox(0,0)[cc]{$=$}} \put(85.00,0.00){\line(-1,4){10.00}} \put(75.00,40.00){\line(1,4){9.92}} \put(84.92,79.67){\line(-3,-4){29.75}} \put(55.17,40.00){\line(3,-4){30.00}} \put(85.17,0.00){\line(-5,4){50.17}} \put(35.00,40.13){\line(5,4){50.00}} \put(85.00,0.00){\line(-5,3){67.00}} \put(18.00,40.20){\line(5,3){67.00}} \put(75.00,40.00){\circle*{3.33}} \put(55.00,40.00){\circle*{3.33}} \put(35.00,40.00){\circle*{3.33}} \put(17.67,40.00){\circle*{3.33}} \end{picture} \end{center} \caption{\label{f-hd-mon} Hasse diagram of $n$ spin one-half state propositional systems $L(x^i), i=1,\cdots ,n$ which are noncomeasurable. The superscript $i$ represents the $i$th measurement direction. } \end{figure} \begin{figure}[htd] \begin{center} %TexCad Options %\grade{\off} %\emlines{\off} %\beziermacro{\off} %\reduce{\on} %\snapping{\off} %\quality{0.20} %\graddiff{0.01} %\snapasp{1} %\zoom{1.00} \unitlength 0.50mm \linethickness{0.4pt} \begin{picture}(151.06,45.00) \put(0.00,35.00){\circle{2.11}} \put(60.00,35.00){\circle{2.11}} \put(0.00,30.00){\makebox(0,0)[cc]{$p_-^1$}} \put(60.00,30.00){\makebox(0,0)[cc]{$p_+^1$}} \put(0.00,35.00){\line(1,0){60.00}} \put(70.00,35.00){\circle{2.11}} \put(130.00,35.00){\circle{2.11}} \put(70.00,30.00){\makebox(0,0)[cc]{$p_-^2$}} \put(130.00,30.00){\makebox(0,0)[cc]{$p_+^2$}} \put(70.00,35.00){\line(1,0){60.00}} \put(30.00,45.00){\makebox(0,0)[cc]{$L(x^1)$}} \put(100.00,45.00){\makebox(0,0)[cc]{$L(x^2)$}} \put(10.00,5.00){\circle{2.11}} \put(70.00,5.00){\circle{2.11}} \put(10.00,0.00){\makebox(0,0)[cc]{$p_-^3$}} \put(70.00,0.00){\makebox(0,0)[cc]{$p_+^3$}} \put(10.00,5.00){\line(1,0){60.00}} \put(90.00,5.00){\circle{2.11}} \put(150.00,5.00){\circle{2.11}} \put(90.00,0.00){\makebox(0,0)[cc]{$p_-^n$}} \put(150.00,0.00){\makebox(0,0)[cc]{$p_+^n$}} \put(90.00,5.00){\line(1,0){60.00}} \put(40.00,15.00){\makebox(0,0)[cc]{$L(x^3)$}} \put(120.00,15.00){\makebox(0,0)[cc]{$L(x^n)$}} \put(80.00,4.67){\makebox(0,0)[cc]{$\cdots$}} \end{picture} \end{center} \caption{\label{f-gd-mon} Greechie diagram of $n$ spin one-half state propositional systems $L(x^i), i=1,\cdots ,n$ which are noncomeasurable. The superscript $i$ represents the $i$th measurement direction. } \end{figure} \section{Finite subalgebras of $n$--dimensional Hilbert logics} \label{lx-xxslfdhl2} The finite subalgebras of two--dimensional Hilbert space are $MO_n, n\in {\Bbb N}$. This can be visualized easily, since given a vector $v$ associated with a proposition $p_v$, there exists only a {\em single} orthogonal vector $v'$, corresponding to the proposition $p_v'$, which is the negation of the proposition $p_v$. Conversely, the negation of $p_v'$ can be uniquely identified with the vector $v$. This is not the case in three- and higher-dimensional spaces, where the complement of a vector is a plane (or a higher-dimensional subspace), and is therefore no unique vector. The previous results can be generalized to $n$--dimensional Hilbert spaces. Take, as an incomplete example, the product of a Boolean algebra of dimension $n-2$ and a modular lattice of the Chinese lantern type $MO_m$ \cite{kalmbach-priv}; e.g., \begin{equation} B\times MO_n=2^{n-2}\times MO_m,\quad 1< m\in {\Bbb N}, \quad n\ge 3. \label{l-nncsu} \end{equation} It is depicted in Figure \ref{f-2n-1xmo2}. \begin{figure} \begin{center} %TexCad Options %\grade{\off} %\emlines{\off} %\beziermacro{\off} %\reduce{\on} %\snapping{\off} %\quality{2.00} %\graddiff{0.01} %\snapasp{1} %\zoom{2.08} \unitlength 1.00mm \linethickness{0.4pt} \begin{picture}(77.13,44.45) \put(0.00,25.00){\circle{2.11}} \put(10.00,25.00){\circle{2.11}} \put(20.00,25.00){\circle{2.11}} \put(30.00,25.00){\circle{2.11}} \put(39.66,25.00){\circle{2.11}} \put(50.00,25.00){\circle{2.11}} \put(65.90,16.80){\circle{2.11}} \put(75.90,6.80){\circle{2.11}} \put(66.07,33.39){\circle{2.11}} \put(76.07,43.39){\circle{2.11}} \put(1.10,25.08){\line(1,0){7.83}} \put(10.97,25.08){\line(1,0){7.83}} \put(20.84,25.08){\line(1,0){7.83}} \put(40.74,25.08){\line(1,0){7.83}} \put(66.89,34.02){\line(1,1){8.51}} \put(66.72,16.16){\line(1,-1){8.51}} \put(32.92,24.91){\circle*{0.68}} \put(34.96,24.91){\circle*{0.68}} \put(37.00,24.91){\circle*{0.68}} \put(70.01,25.08){\circle*{0.68}} \put(68.85,28.85){\circle*{0.68}} \put(68.85,21.13){\circle*{0.68}} \put(24.00,15.04){\makebox(0,0)[cc] {$\underbrace{\qquad \qquad \qquad \qquad \qquad \qquad}_{2^{n-2}}$}} \put(64.67,0.37){\makebox(0,0)[cc]{$\underbrace{\qquad \qquad}_{MO_n}$}} %\bezier{68}(51.08,24.92)(56.53,24.60)(65.17,32.45) \put(51.08,24.92){\line(1,0){1.67}} \multiput(52.75,25.00)(0.45,0.11){4}{\line(1,0){0.45}} \multiput(54.56,25.44)(0.28,0.11){7}{\line(1,0){0.28}} \multiput(56.51,26.23)(0.21,0.11){10}{\line(1,0){0.21}} \multiput(58.60,27.37)(0.17,0.12){13}{\line(1,0){0.17}} \multiput(60.82,28.87)(0.15,0.12){30}{\line(1,0){0.15}} %\end %\bezier{68}(51.08,24.92)(56.53,25.24)(65.17,17.40) \put(51.08,24.92){\line(1,0){1.67}} \multiput(52.75,24.84)(0.45,-0.11){4}{\line(1,0){0.45}} \multiput(54.56,24.40)(0.28,-0.11){7}{\line(1,0){0.28}} \multiput(56.51,23.61)(0.21,-0.11){10}{\line(1,0){0.21}} \multiput(58.60,22.47)(0.17,-0.11){13}{\line(1,0){0.17}} \multiput(60.82,20.98)(0.15,-0.12){30}{\line(1,0){0.15}} %\end \end{picture} \end{center} \caption{\label{f-2n-1xmo2} Finite subalgebra of $n$--dimensional Hilbert logic $2^{n-2}\times MO_m$, $13$. Consider, for instance, four -dimensional real Hilbert space ${\Bbb R}^4$. The product $MO_2\times MO_2$ is a subalgebra of the corresponding Hibert logic but is not a logic represented by Equation (\ref{l-nncsu}). This can be demonstrated by identifying the following eight one--dimensional subspaces\footnote{ $\sp$ denotes the linear span.} \begin{eqnarray*} &a=\sp (1,0,0,0),\quad a'=\sp (0,1,0,0),\quad b=\sp (1,1,0,0),\quad b'=\sp (1,-1,0,0),&\\ &c=\sp (0,0,1,0),\quad c'=\sp (0,0,0,1),\quad d=\sp (0,0,1,1),\quad d'=\sp (0,0,1,-1)& \end{eqnarray*} with the atoms of the two factors $MO_2$ (four atoms per factor) \cite{harding-priv}. Note that $\{a,a',c,c'\}$ and $\{b,b',d,d'\}$ are two orthogonal tripods in ${\Bbb R}^4$. This result could be generalized to the product $MO_i \times MO_j$ by augmenting the above vectors with additional vectors $(\cos \phi_l,\sin \phi_l,0,0)$, $(\sin \phi_l,-\cos \phi_l,0,0)$, $(0,0,\cos \phi_k,\sin \phi_k)$, $(0,0,\sin \phi_k,-\cos \phi_k)$, such that all angles $\phi_{l,k}$ are mutually different, $l,k\in {\Bbb N}$, and $10$, then there exist in ${\Bbb R}^n$ pairwise orthogonal subspaces $M_1,\dots,M_k$ such that ${\Bbb R}^n = M_1 \vee \dots \vee M_k$ and $dim M_{n_i} = n_i$. Thus ${\frak C} ({\Bbb R}^{n_1})\times \dots \times {\frak C} ({\Bbb R}^{n_k})$ is a sub-orthomodular lattice of ${\frak C} ({\Bbb R}^n)$. \item Any sub-orthomodular lattice of a sub-orthomodular lattice is a sub-orthomodular lattice. \item $2^1=\{0 ,1\}$ is a sub-orthomodular lattice of ${\frak C} ({\Bbb R}^n)$ if $n>0$ and $2^p $ is a sub-orthomodular lattice of ${\frak C} ({\Bbb R}^n)$ iff $p\leq n$. \item Any $MO_p$ is a sub-orthomodular lattice of ${\frak C} ({\Bbb R}^{2n})$ but is not a sub-orthomodular lattice of ${\frak C} ({\Bbb R}^{2n+1})$. The reason is: In $MO_p$, 1 is a commutator. 1 is not a commutator in ${\frak C} ({\Bbb R}^{2n+1})$ \cite{cheval-or}. For the same reason, a product of $MO_p$, without a Boolean factor, is not a sub-orthomodular lattice of ${\frak C} ({\Bbb R}^{2n+1})$. \end{itemize} Let $n>0$ be an integer. The finite sub-orthomodular posets of ${\frak C} ({\Bbb R}^n)$ are the $$2^q \times MO_{n_1} \times\dots \times MO_{n_k}\quad \textrm{with}\quad q +2k\leq n.$$ If $n$ is odd then $q$ must be different from 0 (the product must contain a Boolean factor) \cite{cheval-priv}.\footnote{ Notice that $MO_2\times MO_2 $ is, for example, not a sub-orthomodular poset of ${\frak C} ({\Bbb R}^5)$.} Members of this class also have a separating set of two-valued states. Any finite modular orthomodular lattice is isomorphic to a sub-orthomodular lattice of some ${\frak C} ({\Bbb R}^n)$. %the class of all finite modular %ortholattices %$$B\times MO_{m_1}\times MO_{m_2}\times MO_{m_3}\times \cdots \times %MO_{m_k},$$ where $B$ is some Boolean algebra and %$m_1,m_2,m_3,\ldots m_k\in {\Bbb N}-1$ %coincides with the class of all finite subalgebras of %$n$--dimensional Hilbert logics. \section{Probabilities} \subsection{Framework} Let us first programmatically state the framework in which we wish to operate. In our view, probabilities are quantitative expectations of experimental frequencies of certain observed events. The most fundamental yes-no event or outcome in quantum mechanics is a detector click: there either is such a click, or there is none. These events are all that ``we have;'' there is no other empirical physical entity or property, such as human intuition, which can be justifiable called ``physical.'' There is no such thing as certainty in physics (and life in general ;-); thus all physical experience inevitably is stochastic. Induction is ``bottom-up.'' It attempts to reconstruct certain postulated (quantum) structures from such elementary events. The induction problem, in particular effective algorithmic ways and methods to derive certain outcomes or events from other (causally ``previous'') events or outcomes via some kind of ``narratives'' such as physical theories, still remains unsolved. Indeed, in view of powerful formal incompleteness theorems, such as the halting problem, the busy beaver function, or the recursive unsolvability of the rule inference problem, the induction problem is provable recursively unsolvable for physical systems which can be reduced to, or at least contain, universal Turing machines. The physical universe as we know it, appears to be of that kind (cf. Refs.~\cite{svozil-93,svozil-unev}). Deduction is ``top-down.'' It postulates certain entities such as physical theories. Those theories may just have been provided by an oracle, they may be guesswork or just random pieces of data crap in a computer memory. Deduction then derives empirical consequences from those theories. In both cases, probabilities are the only interface between physical theories and the richness of physical experiences which ultimately seem to consist of elementary yes-no events or outcomes. \subsection{Classical probabilities} Classical probabilities are probabilities about events or outcomes of classical systems. Classical systems are Boolean; e.g., distributive, by definition. In what follows, the terms valuation, two-valued (probability) measure, as well as dispersionless state and classical truth assignment will be used as synonyms. Suppose a classical Boolean algebra has $n$ elements. Then there exist $2^n$ such truth assignments. All classical probabilities can be represented as a convex sums of all the possible truth assignments; this is a necessary and sufficient condition. As a consequence, there exist certain constraints on classical probabilities; constraints which already were considered by Boole \cite{Boole,Boole-62} over 150 years ago. In order to establish bounds on quantum probabilities, Pitowsky (among others; see, e.g., Froissart \cite{froissart-81} and Tsirelson \cite{cirelson:80,cirelson}) has given a geometrical interpretation of the bounds of classical probabilities in terms of correlation polytopes \cite{pitowsky-86,pitowsky,pitowsky-89a,Pit-91,Pit-94}. Consider an arbitrary number of classical events or outcomes $a_1, a_2,\ldots , a_n$. Take some (or all of) their probabilities and some (or all of) the joint probabilities $p_1, p_2,\ldots , p_n, p_{12},\ldots $ and identify them with the components of a vector $p=(p_1, p_2,\ldots , p_n, p_{12},\ldots )$ formed in Euclidean space. Suppose that the events or outcomes $a_1, a_2,\ldots , a_n$ are independent. Then the probabilities $p_i$, $i=1,\ldots ,n$ may acquire the extreme cases $0,1$ independently. Consider all vectors spanned by those ``extremal'' components. The combined values of $p_1, p_2,\ldots , p_n$ of the extreme cases $p_i=0,1$, together with the joined probabilities $p_{ij} =p_i p_j$ can also be interpreted as rows of a truth table; with $0,1$ corresponding to ``{\it false}'' and ``{\it true,}'' respectively. Moreover, any such entry corresponds to a {\em two-valued measure} (also called {\em valuation, 0-1-measure} or {\em dispersionless measure}). In geometrical terms, any classical probability distribution is representable by some convex sum over all two-valued measures characterized by the row entries of the truth tables. That is, it corresponds to some point on the face of the classical correlation polytope $C={\rm conv} (K)$ which is defined by the set of all points whose convex sum extends over all vectors associated with row entries in the truth table $K$. More precisely, consider the convex hull ${\rm conv} (K)=\left\{ \sum_{i=1}^{2^n} \lambda_i{\bf x}_i \; \left| \; \lambda_i\ge 0,\; \sum_{i=1}^{2^n}\lambda_i =1 \right. \right\} $ of the set $$K =\{{\bf x}_1,{\bf x}_2,\ldots ,{\bf x}_{2^n}\} = \left\{ \left. \large(t_1, t_2,\ldots , t_n, t_xt_y,\ldots \large) \; \right| \; t_i \in \{0,1\},\; i=1,\ldots ,n \right\}.$$ Here, the terms $t_xt_y,\ldots$ stand for arbitrary products associated with the joint propositions which are considered. Exactly what terms are considered depends on the particular physical configuration. By the Minkoswki-Weyl representation theorem \cite[p.29]{ziegler}, every convex polytope has a dual (equivalent) description: (i) either as the convex hull of its extreme points; i.e., vertices; (ii) or as the intersection of a finite number of half-spaces, each one given by a linear inequality. The linear inequalities, which are obtained from the set $K$ of vertices by solving the so called {\em hull problem} coincide with Boole's ``conditions of possible experience.'' For particular physical setups, the inequalities can be identified with Bell-type inequalities which have to be satisfied by all classical probability distributions. These conditions are demarcation criteria; i.e., they are complete and maximal in the sense that no other system of inequality exist which characterizes the correlation polytopes completely and exhaustively. (That is, the bounds on probabilities cannot be enlarged and improved). Generalizations to the joint distributions of more than two particles are straightforward. Correlation polytopes have provided a systematic, constructive way of finding the entire set of Bell-type inequalities associated with any particular physical configuration \cite{2000-poly,2001-cddif}, although from a computational complexity point of view \cite{garey}, the problem remains intractable \cite{Pit-91}. \subsection{Quantum probabilities} Quantum logic suggests that the classical Boolean propositional structure of events should be replaced by the Hilbert lattice $\ll$ of subspaces of a Hilbert space $\hh$. (Alternatively, we may use the set of all projection operators $\ppp$.) Thus we should be able to define a probability measure on subspaces of a Hilbert space as a normed function $P$ which assigns to every subspace a nonnegative real number such that if $\{\mm_{p_i} \}$ is any countable set of mutually orthogonal subspaces (corresponding to comeasurable propositions $p_i$) having closed linear span $\mm_{\vee_i p_i}=\oplus_i \mm_{p_i}$, then \begin{equation} P(\mm_{\vee_i p_i})=\sum_i P(\mm_{p_i}). \label{l-ka1} \end{equation} Furthermore, the tautology corresponding to the entire Hilbert space should have probability one. That is, \begin{equation} P({\frak H})=1. \label{l-ka2} \end{equation} Instead of the subspaces, we could have considered the corresponding projection operators. A measure of the above type can be obtained by selecting an arbitrary normalized vector $ y\in \hh$ and by identifying $P_{y}(\mm_x)$ with the square of the absolute value of the scalar product of the orthogonal projection of $y$ onto $\mm_x$ spanned by the unit vector $x$, $$P_{ y}(\mm_x) =\vert (x,y)\vert^2 .$$ More generally, any linear combination $\sum_i P_{ x_i}$ of such measures $P_{ x_i}$ is again such a measure. In what follows we shall refer to states corresponding to one--dimensional subspaces or projections as {\em pure states}. \index{pure state} Indeed, a celebrated theorem by Gleason \cite{Gleason} states that in a separable Hilbert space of dimension at least three, {\em every} probability measure on the projections satisfying (\ref{l-ka1}) and (\ref{l-ka2}) can be written in the form \begin{equation} \label{l-gleason} P_\rho (E_\mm )=\textrm{trace}(\rho E_\mm). \end{equation} $E_\mm$ denotes the orthogonal projection on $\mm$ and $\rho $ is a unique positive (semi-definite) self-adjoint {\em density operator} of the trace class; \index{density operator} i.e., $(\rho x,x)=(x,\rho x)\ge 0$ for all $x\in \hh$, and $\textrm{trace}(\rho )=1$. Then the {\em expectation value} \index{expectation value} of an observable corresponding to a self-adjoint operator $A$ with eigenvalues $\lambda_i$ is (in $n$--dimensional Hilbert space) given by \begin{equation} \label{l-gleason1} \langle A\rangle = \sum_{i=1}^n \lambda_i P(E_i) = \sum_{i=1}^n \lambda_i \textrm{trace}(\rho E_i)= \textrm{trace}(\rho A). \end{equation} The relationship between (\ref{l-gleason}) and (\ref{l-gleason1}) is due to the spectral decomposition of $A$ . Gleason's theorem can be seen as a {\it substitute} for the probability axiom of quantum mechanics by deriving it from some ``fundamental'' assumptions and ``reasonable'' requirements. One such requirement is that, if $E_p$ and $E_q$ are orthogonal projectors representing comeasurable, independent propositions $p$ and $q$, then their join $p\vee q$ (corresponding to $E_p+E_q$) has probability $P(p\vee q)=P(p)+P(q)$ (corresponding to $P(E_p+E_q)=P(E_p)+P(E_q)$). \section{Contextuality} In the late 50's, Ernst Specker was considering the question of whether it might be possible to consistently define elements of physical reality {\em ``globally''} which can merely be measured {\em ``locally''} \cite{specker-60}. Specker mentions the scholastic speculation of the so-called ``infuturabilities''; that is, the question of {\em whether \index{infuturabilities} the omniscience (comprehensive knowledge) of God extends to events which would have occurred if something had happened which did not happen} (cf. \cite[p. 243]{specker-60} and \cite[p. 179]{specker-ges}). Today, the scholastic term ``infuturability'' would be called ``counterfactual.'' Let us be more specific. Here, the meaning of the terms local and global will be understood as follows. In quantum mechanics, every single orthonormal basis of a Hilbert space corresponds to locally comeasurable elements of physical reality. The (undenumerable) class of all orthonormal basis of a Hilbert space corresponds to a global description of the conceivable observables --- Schr\"odinger's catalogue of expectation values \cite{schrodinger}. It is quite reasonable to ask whether one could (re)construct the global description from its single, local, parts, whether the pieces could be used to consistently define the whole. A metaphor of this motive is the quantum jigsaw puzzle depicted in Figure \ref{f-jigsaw}. In this jigsaw puzzle, all legs should be translated to the origin. Every single piece of the jigsaw puzzle consists of mutually orthogonal rays. It has exactly one ``privileged'' leg, which is singled out by coloring it differently from the other (mutual) orthogonal legs (or, alternatively, assigning to it the probability measure one, corresponding to certainty). The pieces should be arranged such that one and the same leg occurring in two or more pieces should have the same color (probability measure) for every piece. \begin{figure} \begin{center} %TexCad Options %\grade{\off} %\emlines{\off} %\beziermacro{\off} %\reduce{\on} %\snapping{\off} %\quality{2.00} %\graddiff{0.01} %\snapasp{1} %\zoom{7.45} \unitlength 0.80mm \linethickness{0.4pt} \begin{picture}(107.62,83.81) \put(15.48,83.81){\line(1,-1){10.00}} \put(25.48,73.81){\line(-1,-3){5.00}} \put(20.48,58.81){\line(-3,-2){15.00}} \put(5.48,48.81){\line(-1,3){5.00}} \put(0.48,63.81){\line(1,3){5.00}} \put(5.48,78.81){\line(2,1){10.00}} \put(15.48,73.81){\line(-1,-2){5.00}} \put(10.48,63.81){\line(1,-1){5.00}} \put(10.48,63.81){\line(-1,0){5.00}} \put(84.05,15.24){\line(-1,1){10.00}} \put(74.05,25.24){\line(-3,-1){15.00}} \put(59.05,20.24){\line(-2,-3){10.00}} \put(49.05,5.24){\line(3,-1){15.00}} \put(64.05,0.24){\line(3,1){15.00}} \put(79.05,5.24){\line(1,2){5.00}} \put(74.05,15.24){\line(-2,-1){10.00}} \put(64.05,10.24){\line(-1,1){5.00}} \put(64.05,10.24){\line(0,-1){5.00}} \put(50.48,73.81){\line(-1,-2){5.00}} \put(45.48,63.81){\line(2,-1){20.00}} \put(65.48,53.81){\line(-1,3){5.00}} \put(60.48,68.81){\line(3,1){15.00}} \put(75.48,73.81){\line(-3,2){15.00}} \put(60.48,83.81){\line(-1,0){10.00}} \put(50.48,83.81){\line(0,-1){10.00}} \put(55.48,78.81){\line(0,-1){10.00}} \put(55.48,68.81){\line(1,-1){5.00}} \put(55.48,68.81){\line(-1,-1){5.00}} \put(33.81,14.29){\line(1,2){5.00}} \put(38.81,24.29){\line(-2,1){20.00}} \put(18.81,34.29){\line(1,-3){5.00}} \put(23.81,19.29){\line(-3,-1){15.00}} \put(8.81,14.29){\line(3,-2){15.00}} \put(23.81,4.29){\line(1,0){10.00}} \put(33.81,4.29){\line(0,1){10.00}} \put(28.81,9.29){\line(0,1){10.00}} \put(28.81,19.29){\line(-1,1){5.00}} \put(28.81,19.29){\line(1,1){5.00}} \put(30.48,53.81){\line(-1,-3){5.00}} \put(25.48,38.81){\line(5,-1){25.00}} \put(50.48,33.81){\line(1,1){10.00}} \put(60.48,43.81){\line(-4,3){20.00}} \put(40.48,58.81){\line(-2,-1){10.00}} \put(35.48,48.81){\line(2,-1){10.00}} \put(45.48,43.81){\line(1,1){5.00}} \put(45.48,43.81){\line(0,-1){5.00}} \put(10.71,63.81){\line(1,2){5.00}} \put(10.24,63.81){\line(1,2){5.00}} \put(45.48,43.57){\line(-2,1){10.00}} \put(45.48,44.05){\line(-2,1){10.00}} \put(55.00,68.57){\line(-1,-1){4.76}} \put(50.71,63.81){\line(1,1){4.76}} \put(64.20,10.04){\line(2,1){10.24}} \put(64.05,10.48){\line(2,1){10.00}} \put(28.57,19.52){\line(0,-1){10.71}} \put(29.05,8.81){\line(0,1){10.71}} \put(100.00,50.00){\line(0,-1){10.00}} \put(100.00,40.00){\line(5,-2){7.62}} \put(100.00,40.02){\line(-5,-3){5.95}} \put(98.32,33.33){\line(1,4){1.67}} \put(99.98,40.00){\line(-2,1){5.71}} \put(97.38,45.24){\line(1,-2){2.62}} \put(100.00,40.00){\line(2,1){5.24}} \put(75.00,40.00){\vector(1,0){10.00}} \put(80.00,45.00){\makebox(0,0)[cc]{?}} \put(37.14,72.86){\makebox(0,0)[cc]{$\cdots$}} \put(50.00,23.81){\makebox(0,0)[cc]{$\cdots$}} \put(74.05,61.90){\makebox(0,0)[cc]{$\cdots$}} \put(6.67,30.71){\makebox(0,0)[cc]{$\cdots$}} \put(102.86,35.24){\makebox(0,0)[cc]{$\cdots$}} \put(103.33,46.19){\makebox(0,0)[cc]{$\ddots$}} \end{picture} \end{center} \caption{\label{f-jigsaw} The quantum jigsaw puzzle in three dimensions: is it possible to consistently arrange undenumerably many pieces of counterfactual observables, only one of which is actually measurable? Every tripod has a red leg (thick line) and two green legs.} \end{figure} As it turns out, for Hilbert spaces of dimension greater than two, the jigsaw puzzle is unsolvable. That is, every attempt to arrange the pieces consistently into a whole is doomed to fail. One characteristic of this failure is that {\em legs (corresponding to elementary propositions) appear differently colored, depending on the particular tripod they are in!} More explicitly: there may exist two tripods (embedded in a larger tripod set) with one common leg, such that this leg appears red in one tripod and green in the other one. Since every tripod is associated with a system of mutually compatible observables, this could be interpreted as an indication that the truth or falsity of a proposition (and hence the element of physical reality) associated with it depends on the {\em context} of measurement \cite{bell-66,redhead} \index{context dependence}; i.e., whether it is measured along with first or second frame of mutually compatible observables.\footnote{Measurement of propositions corresponding to a given triad can be reduced to a single ``Ur''-observable per triad.} It is in this sense that the nonexistence of two-valued probability measures is a formalization of the concept of context dependence or contextuality. \index{contextuality} Observe that at this point, the theory takes an unexpected turn. The whole issue of a ``secret classical arena beyond quantum mechanics'', more specifically noncontextual hidden parameters, boils down to a consistent coloring of a finite number of vectors in three--dimensional space! One of the most compact and comprehensive versions of the Kochen-Specker argument in three--dimensional Hilbert space ${\Bbb R}^3$ has been given by Peres \cite{peres-91}. (For other discussions, see Refs. \cite{stairs83,redhead,jammer-92,brown,peres-91,peres,penrose-ks,clifton-93,mermin-93,svozil-tkadlec}.) Peres' version uses a 33-element set of lines without any two-valued state. The direction vectors of these lines arise by all permutations of coordinates from \begin{equation} (0,0,1),\; (0,\pm 1,1),\; (0,\pm 1,\sqrt{2}),\; \;\textrm{ and }\; (\pm 1,\pm 1,\sqrt{2}). \label{p-ksl} \end{equation} As will be explicitly enumerated below, these lines can be generated (by the {\it nor}-operation between nonorthogonal propositions) by the three lines \cite{svozil-tkadlec} $$(1,0,0),\; (1,1,0),\; (\sqrt2,1,1).$$ Note that as three arbitrary but mutually nonorthogonal lines generate a dense set of lines (cf. \cite{havlicek}), it can be expected that any such triple of lines (not just the one explicitly mentioned) generates a finite set of lines which does not allow a two-valued probability measure. The way it is defined, this set of lines is invariant under interchanges (permutations) of the $x_1,x_2$ and $x_3$ axes, and under a reversal of the direction of each of these axes. This symmetry property allows us to assign the probability measure $1$ to some of the rays without loss of generality --- assignment of probability measure $0$ to these rays would be equivalent to renaming the axes, or reversing one of the axes. The Greechie diagram of the Peres configuration is given in Figure~\ref{sk33} \cite{svozil-tkadlec}. For simplicity, 24~points which belong to exactly one edge are omitted. The coordinates should be read as follows: $\bar{1}\rightarrow -1$ and $2\rightarrow \sqrt{2}$; e.g., 1$\bar{1}$2 denotes %%%%%%%%%%%%%%%%%%%%%%%% $\textrm{Sp} (1,-1,\sqrt{2})$. Concentric circles indicate the (non orthogonal) generators mentioned above. \newcommand{\emline}[4]% {\put(#1,#2){\special{em:moveto}}\put(#3,#4){\special{em:lineto}}} \newsavebox{\vertex}\savebox{\vertex}{% % vertex in diagrams {\unitlength1mm\begin{picture}(0,0)\put(0,0){\circle{1}}\end{picture}}} \begin{figure}[p] \unitlength .45\textwidth \newsavebox{\subdiagram}\savebox{\subdiagram}{% % circle in diagrams {\unitlength1mm\begin{picture}(0,0)\put(0,0){\circle{2}}\end{picture}}} \newcommand{\discbig}{\usebox{\subdiagram}} \newcommand{\emlin}[4]% {\put(#1,#2){\special{em:moveto}}\put(#3,#4){\special{em:lineto}}} \newcommand{\disc}{\usebox{\vertex}} \newcommand{\place}[6]% {\put(#1,#2){\hspace{#3pt}\raisebox{#4pt}{\makebox(0,0)[#5]{$#6$}}}} \newcommand{\point}[2]{\put(#1,#2){\disc}} \newcommand{\onethird}[3]{\axis#1\side#1#2\side#1#3\cross#2#3} \newcommand{\axis}[4]% {\point#1\point#2\point#3\point#4\emlin#1#3\emlin#3{0}{0}} \newcommand{\side}[8]% {\point#5\point#6\point#7\point#8\emlin#1#5\emlin#5#6\emlin#2#8\emlin#8#7} \newcommand{\cross}[8]{\emlin#1#7\emlin#7#3\emlin#3#5\emlin#2{0}{0}} \catcode`\!=\active \def!{\bar1} \begin{center} \begin{picture}(2,2)(-1,-1) \onethird {{{ 0.100}{-0.995}}{{ 0.050}{-0.747}}{{ 0.000}{-0.500}}{{ 0.000}{-0.250}}}% {{{ 0.643}{-0.766}}{{ 0.087}{-0.050}}{{ 0.100}{-0.250}}{{ 0.536}{-0.112}}}% {{{-0.643}{-0.766}}{{-0.087}{-0.050}}{{-0.100}{-0.250}}{{-0.536}{-0.112}}}% \onethird {{{ 0.812}{ 0.584}}{{ 0.622}{ 0.417}}{{ 0.433}{ 0.250}}{{ 0.217}{ 0.125}}}% {{{ 0.342}{ 0.940}}{{-0.000}{ 0.100}}{{ 0.167}{ 0.212}}{{-0.171}{ 0.520}}}% {{{ 0.985}{-0.174}}{{ 0.087}{-0.050}}{{ 0.267}{ 0.038}}{{ 0.365}{-0.408}}}% \onethird {{{-0.912}{ 0.411}}{{-0.672}{ 0.330}}{{-0.433}{ 0.250}}{{-0.217}{ 0.125}}}% {{{-0.985}{-0.174}}{{-0.087}{-0.050}}{{-0.267}{ 0.038}}{{-0.365}{-0.408}}}% {{{-0.342}{ 0.940}}{{-0.000}{ 0.100}}{{-0.167}{ 0.212}}{{ 0.171}{ 0.520}}}% \put (0,0){\circle{0.2}} \put ( 0.000, 0.100){\discbig} \put ( 0.050,-0.747){\discbig} \put ( 0.217, 0.125){\discbig} \footnotesize \place { 0.100}{-0.995}{ 0}{-6}{ t}{2!!} \place { 0.050}{-0.747}{ 6}{ 0}{l }{211} \place { 0.000}{-0.500}{ 6}{ 0}{l }{01!} \place { 0.000}{-0.250}{ 2}{ 3}{lb}{011} \place { 0.000}{ 0.100}{ 0}{-4}{ t}{100} \place { 0.100}{-0.250}{ 4}{-4}{lt}{2!1} \place {-0.100}{-0.250}{-4}{-4}{rt}{21!} \place { 0.643}{-0.766}{ 6}{ 0}{l }{102} \place { 0.365}{-0.408}{ 6}{ 0}{l }{20!} \place { 0.087}{-0.050}{ 4}{ 0}{lb}{010} \place {-0.643}{-0.766}{-6}{ 0}{r }{120} \place {-0.365}{-0.408}{-6}{ 0}{r }{2!0} \place {-0.087}{-0.050}{-4}{ 0}{rb}{001} \place { 0.812}{ 0.584}{ 6}{ 0}{lb}{!!2} \place { 0.622}{ 0.417}{ 6}{ 0}{lt}{112} \place {-0.912}{ 0.411}{-6}{ 0}{rb}{!2!} \place {-0.672}{ 0.330}{-6}{ 0}{rt}{121} \place { 0.433}{ 0.250}{ 3}{-3}{lt}{1!0} \place { 0.217}{ 0.125}{-6}{ 1}{r }{110} \place {-0.433}{ 0.250}{-3}{-3}{rt}{10!} \place {-0.217}{ 0.125}{ 6}{ 1}{l }{101} \place { 0.267}{ 0.038}{ 6}{ 0}{l }{1!2} \place { 0.167}{ 0.212}{ 0}{ 6}{lb}{!12} \place {-0.267}{ 0.038}{-6}{ 0}{r }{12!} \place {-0.167}{ 0.212}{ 0}{ 6}{rb}{!21} \place { 0.985}{-0.174}{ 4}{-6}{ t}{201} \place { 0.536}{-0.112}{ 0}{-6}{lt}{!02} \place {-0.985}{-0.174}{-4}{-6}{ t}{210} \place {-0.536}{-0.112}{ 0}{-6}{rt}{!20} \place { 0.342}{ 0.940}{ 0}{ 6}{ b}{021} \place { 0.171}{ 0.520}{-4}{ 4}{rb}{0!2} \place {-0.342}{ 0.940}{ 0}{ 6}{ b}{012} \place {-0.171}{ 0.520}{ 4}{ 4}{lb}{02!} \end{picture} \end{center} \caption{Greechie diagram of a set of propositions embeddable in ${\Bbb R}^3$ without any two-valued probability measure \protect\cite[Figure 9]{svozil-tkadlec}. \label{sk33}} \end{figure} Let us prove that there is no two-valued probability measure \cite{svozil-tkadlec,tkadlec-96}. Due to the symmetry of the problem, we can choose a particular coordinate axis such that, without loss of generality, $P(100)=1$. Furthermore, we may assume (case 1) that $P(21\bar{1}) = 1$. It immediately follows that $P(001) = P(010) = P(102) = P(\bar{1}20) = 0$. A second glance shows that $P(20\bar{1}) = 1$, $P(1\bar{1}2) = P(112) = 0$. Let us now suppose (case 1a) that $P(201) = 1$. Then we obtain $P(\bar{1}12) = P(\bar{1}\bar{1}2) = 0$. We are forced to accept $P(110) = P(1\bar{1}0) = 1$ --- a contradiction, since $(110)$ and $(1\bar{1}0)$ are orthogonal to each other and lie on one edge. Hence we have to assume (case 1b) that $P(201) = 0$. This gives immediately $P(\bar{1}02)=1$ and $P(211) =0$. Since $P(01\bar{1})=0$, we obtain $P(2\bar{1}\bar{1})=1$ and thus $P(120)=0$. This requires $P(2\bar{1}0)=1$ and therefore $P(12\bar{1})=P(121)=0$. Observe that $P(210) = 1$, and thus $P(\bar{1}2\bar{1}) = P(\bar{1}21) = 0$. In the following step, we notice that $P(10\bar{1}) = P(101) = 1$ --- a contradiction, since $(101)$ and $(10\bar{1})$ are orthogonal to each other and lie on one edge. Thus we are forced to assume (case 2) that $P(2\bar{1}1) = 1$. There is no third alternative, since $P(011)=0$ due to the orthogonality with $(100)$. Now we can repeat the argument for case 1 in its mirrored form. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% The above mentioned set of lines (\ref{p-ksl}) orthogenerates (by the {\it nor}-opera\-tion between orthogonal vectors) a suborthoposet of~${\Bbb R}^3$ with 116~elements; i.e., with 57~atoms corresponding to one--dimensional subspaces spanned by the vectors just mentioned --- the direction vectors of the remaining lines arise by all permutations of coordinates from $(\pm 1,\pm 1,\sqrt{2})$ --- plus their two--dimensional orthogonal planes plus the entire Hilbert space and the null vector \cite{svozil-tkadlec}. This suborthoposet of~${\Bbb R}^3$ has a 17-element set of orthogenerators; i.e; lines with direction vectors $(0,0,1)$, $(0,1,0)$ and all coordinate permutations from $(0,1,\sqrt2)$, $(1,\pm1,\sqrt2)$. It has a 3-element set of generators $$(1,0,0),\; (1,1,0),\; (\sqrt2,1,1).$$ More explicitly, { \let\charmi=A\let\charmii=B % abbr. of coord.: A=-1, B=-2 \newcommand{\ex}[1]% {\let\char#1\ifx\char\charmi-1\else\ifx\char\charmii-2\else#1\fi\fi} \newcommand{\gen}[9]{\textrm{Sp}(\ex#1,\ex#2,\ex#3)&=& (\textrm{Sp}(\ex#4,\ex#5,\ex#6)\oplus \textrm{Sp}(\ex#7,\ex#8,\ex#9))' \equiv \\*&&} \def\sqrtii{\sqrt2}\catcode`\2=\active\def2{\sqrtii} \catcode`\+=\active\def+{\, \textit{nor}\,} \begin {eqnarray*} \textrm{Sp}(1,0,0) &=& a ,\\ \textrm{Sp}(1,1,0) &=& b ,\\ \textrm{Sp}(2,1,1) &=& c ,\\ \gen 001100110 (a+b),\\ \gen 01A100211 (a+c),\\ \gen 010100001 (a+(a+b)),\\ \gen 01110001A (a+(a+c)),\\ \gen 1A0110001 (b+(a+b)),\\ \gen A20211001 (c+(a+b)),\\ \gen 2AA21101A (c+(a+c)),\\ \gen A02211010 (c+(a+(a+b))),\\ \gen 210001A20 ((a+b)+(c+(a+b))),\\ \gen 1200012AA ((a+b)+(c+(a+c))),\\ \gen 1020102AA ((a+(a+b))+(c+(a+c))),\\ \gen 21A011A20 ((a+(a+c))+(c+(a+b))),\\ \gen 201010A02 ((a+(a+b))+(c+(a+(a+b)))),\\ \gen 2A0001120 ((a+b)+((a+b)+(c+(a+c)))),\\ \gen 2A1011A02 ((a+(a+c))+(c+(a+(a+b)))),\\ \gen A12110201 (b+((a+(a+b))+(c+(a+(a+b))))),\\ \gen 02A100A12 (a+(b+((a+(a+b))+\\*% &&(c+(a+(a+b)))))),\\ \gen 20A010102 ((a+(a+b))+((a+(a+b))+\\*% &&(c+(a+c)))),\\ \gen 1A2110A12 (b+(b+((a+(a+b))+\\*% &&(c+(a+(a+b)))))),\\ \gen 01210002A (a+(a+(b+((a+(a+b))+\\*% &&(c+(a+(a+b))))))),\\ \gen 0211001A2 (a+(b+(b+((a+(a+b))+\\*% &&(c+(a+(a+b))))))),\\ \gen AA21A0201 ((b+(a+b))+((a+(a+b))+\\*% &&(c+(a+(a+b))))),\\ \gen 0A2100021 (a+(a+(b+(b+((a+(a+b))+\\*% &&(c+(a+(a+b)))))))),\\ \gen 1121A002A ((b+(a+b))+(a+(b+((a+(a+b))+\\*% && (c+(a+(a+b))))))),\\ \gen A2A210012 (((a+b)+(c+(a+b)))+\\*% &&(a+(a+(b+((a+(a+b))+\\*% &&(c+(a+(a+b)))))))),\\ \gen A212100A2 (((a+b)+(c+(a+b)))+\\*% &&(a+(a+(b+(b+((a+(a+b))+\\*% &&(c+(a+(a+b))))))))),\\ \gen 12A2A0012 (((a+b)+((a+b)+(c+(a+c))))+\\*% &&(a+(a+(b+((a+(a+b))+\\*% &&(c+(a+(a+b)))))))),\\ \gen A01010A2A ((a+(a+b))+(((a+b)+(c+(a+b)))+\\*% &&(a+(a+(b+((a+(a+b))+\\*% &&(c+(a+(a+b))))))))),\\ \gen 1212A00A2 (((a+b)+((a+b)+(c+(a+c))))+\\*% &&(a+(a+(b+(b+((a+(a+b))+\\*% &&(c+(a+(a+b))))))))),\\ \gen 101010A21 ((a+(a+b))+(((a+b)+(c+(a+b)))+\\*% &&(a+(a+(b+(b+((a+(a+b))+\\*% &&(c+(a+(a+b)))))))))). \end{eqnarray*} } So far, we have studied the implosion of the quantum jigsaw puzzle. What about its explosion? What if we try to actually measure the two-valued probability assignments? First of all, we have to clarify what ``measurement'' means. Indeed, in the three--dimensional cases, from all the numerous tripods represented here as lines, only a single one can actually be ``measured'' in a straightforward way. All the others have to be counterfactually inferred. Thus, of course, only if {\em all} propositions --- and not just the ones which are comeasurable --- are counterfactually inferred and compared, we would end up in a complete contradiction. In doing so, we accept the EPR definition of ``element of physical reality.'' As a fall-back option we may be willing to accept that ``actual elements of physical reality'' are determined only by the measurement context. This is not as mindboggling as it first may appear. It should be noted that in finite--dimensional Hilbert spaces, any two {\em commuting} self-adjoint operators $A$ and $B$ corresponding to observables can be simultaneously diagonalized \cite[section 79]{halmos-vs}. Furthermore, $A$ and $B$ commute if and only if there exists a self-adjoint ``Ur''-operator $U$ and two real-valued functions \index{Ur-operator} $f$ and $g$ such that $A=f(U)$ and $B=g(U)$ (cf. \cite[Section 84]{halmos-vs}, Varadarajan \cite[p. 119-120, Theorem 6.9]{varadarajanI} and Pt{\'{a}}k and Pulmannov{\'{a}} \cite[p. 89, Theorem 4.1.7]{pulmannova-91}). A generalization to an arbitrary number of mutually commuting operators is straightforward. Stated pointedly: every set of mutually commuting observables can be represented by just one ``Ur''-operator, such that all the operators are functions thereof. One example is the spin one-half case. There, for instance, the commuting operators are $A={\Bbb I}$ and $B=\sigma_1$ (uncritical factors have been omitted). In this case, take $U=B$ and $f(x)=x^2$, $g(x)=x$. For spin component measurements along the Cartesian coordinate axes $(1,0,0)$, $(0,1,0)$ and $(0,0,1)$, the ``Ur''-operator for the tripods used for the construction of the Kochen-Specker paradox is ($\hbar = 1$)\cite{kochen1} \begin{equation} U=aJ_1^2+bJ_2^2+cJ_3^2= {1\over 2} \left( \begin{array}{ccc} a+b+2c&0&a-b\\ 0&2a+2b&0\\ a-b&0&a+b+2c \end{array} \right), \label{le-ndfuo} \end{equation} where $a,b$ and $c$ are mutually distinct real numbers and $$ J_1^2= {1\over 2} \left( \begin{array}{ccc} 1&0&1\\ 0&2&0\\ 1&0&1 \end{array} \right), \; J_2^2= {1\over 2} \left( \begin{array}{ccc} 1&0&-1\\ 0&2&0\\ -1&0&1 \end{array} \right), \; J_3^2= \left( \begin{array}{ccc} 1&0&0\\ 0&0&0\\ 0&0&1\\ \end{array} \right) $$ are the squares of the spin state observables. Since $J_1^2,J_2^2,J_3^2$ are commuting and all functions of $U$, they can be identified with the observables constituting the tripods. (See also References \cite[pp. 199-200]{peres} and \cite{rzbb,swift80a}.) Let us be a little bit more explicit. We have \begin{eqnarray*} &&J_1^2=[(a-b)(c-a)]^{-1}[U-(b+c)](U-2a),\\ &&J_2^2=[(a-b)(b-c)]^{-1}[U-(a+c)](U-2b),\\ &&J_3^2=[(c-a)(b-c)]^{-1}[U-(a+b)](U-2c). \end{eqnarray*} The diagonal form of the ``Ur''-operator (\ref{le-ndfuo}) is $$ U= \left( \begin{array}{ccc} a+b&0&0\\ 0&b+c&0\\ 0&0&a+c \end{array} \right). $$ Measurement of $U$ can, for instance, be realized by a set of beam splitters \cite{rzbb}; or in an arrangement proposed by Kochen and Specker \cite{kochen1}. Any such measurement will yield either the eigenvalue $a+b$ (exclusive) or the eigenvalue $b+c$ (exclusive) or the eigenvalue $a+c$. Since $a,b,c$ are mutually distinct, one always knows which one of the eigenvalues it is. Furthermore, we observe that $$J_1^2+J_2^2+J_3^2=2{\Bbb I}.$$ Since the possible eigenvalues of any $J_i^2, i=1,2,3$ are either~0 or~1, the eigenvalues of two observables $J_i^2,i=1,2,3$ must be~1, and one must be~0. Any measurement of the ``Ur''-operator $U$ thus yields $a+b$ associated with $J_1^2=J_2^2=1$, $J_3^2=0$ (exclusive) or $a+c$ associated with $J_1^2=J_3^2=1$, $J_2^2=0$ (exclusive) or $b+c$ associated with $J_2^2=J_3^2=1$, $J_1^2=0$. We now consider then the following propositions \begin{description} \item[$p_1$:] The measurement result of $J_1$ is~0, \item[$p_2$:] The measurement result of $J_2$ is~0, \item[$p_3$:] The measurement result of $J_2$ is~0; \end{description} or equivalently, \begin{description} \item[$p_1$:] The measurement result of $U$ is~$b+c$, \item[$p_2$:] The measurement result of $U$ is~$a+c$, \item[$p_3$:] The measurement result of $U$ is~$a+b$. \end{description} For spin component measurements along a different set $\bar x,\bar y,\bar z$ of mutually orthogonal rays, the ``Ur''-operator is given by $$\bar U=\bar a{\bar J}_1^2+\bar b{\bar J}_2^2+\bar c{\bar J}_3^2,$$ where $$ {\bar J}_1=S(\bar x),\quad {\bar J}_2=S(\bar y),\quad {\bar J}_3=S(\bar z).$$ Let us, for example, take $\bar x=(1/\sqrt{2})(1,1,0)$, $\bar y=(1/\sqrt{2})(-1,1,0)$, and $\bar z=z$. In terms of polar coordinates $\theta ,\phi ,r$, these orthogonal directions are $\bar x=(\pi /2,\pi /4,1)$, $\bar y=(\pi /2,-\pi /4,1)$, and $\bar z=(0,0,1)$, and \begin{eqnarray*} {\bar J}_1^2= \left(S({\pi \over 2}, {\pi \over 4})\right)^2&=& {1\over 2} \left( \begin{array}{ccc} 1&0&-i\\ 0&2&0\\ i&0&1 \end{array} \right), \\ {\bar J}_2^2= \left(S({\pi \over 2}, -{\pi \over 4})\right)^2&=& {1\over 2} \left( \begin{array}{ccc} 1&0&i\\ 0&2&0\\ -i&0&1 \end{array} \right), \\ {\bar J}_3^2=\left(S(0,0)\right)^2&=& \left( \begin{array}{ccc} 1&0&0\\ 0&0&0\\ 0&0&1\\ \end{array} \right). \end{eqnarray*} The ``Ur''-operator takes on the matrix form $$ \bar U= {1\over 2} \left( \begin{array}{ccc} {\bar a}+{\bar b}+2{\bar c}&0&-i{\bar a}+i\bar b\\ 0&2{\bar a}+2\bar b&0\\ i{\bar a}-i\bar b&0&{\bar a}+{\bar b}+2{\bar c} \end{array} \right).$$ As expected, the eigenvalues of $\bar U$ are $ \bar a+\bar b, \bar b+\bar c, \bar a+\bar c$. This result holds true for arbitrary rotations $S0(3)$ \cite{murnaghan} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% of the coordinate axes (tripod), parameterized, for instance, by the Euler angles $\alpha ,\beta , \gamma$. Hence, stated pointedly and repeatedly, any measurement of elements of physical reality boils down, in a sense, to measuring a {\em single} ``Ur''-observable, from which the three observables in the tripod can be derived. Different tripods correspond to different ``Ur''-observables. \section{Algebraic options for a ``completion'' of quantum mechanics} Just how far might a classical understanding of quantum mechanics in principle be possible? We shall attempt an answer to this question in terms of mappings of quantum universes into classical ones, more specifically in terms of embeddings of quantum logics into classical logics. We shall also shortly discuss surjective extensions (many-to-one mappings) of classical logics into quantum logics.\footnote{ No attempt will be made here to give a comprehensive review of hidden parameter models. See, for instance, an article by Gudder \cite{gudder1}, where a different approach to the question of hidden parameters is pursued. For a historical review, see the books by Jammer \cite{jammer:66,jammer1}.} It is always possible to enlarge a quantum logic to a classical logic, thereby mapping the quantum logic into the classical logic. In algebraic terms, the question is how much structure can be preserved. A possible ``completion'' of quantum mechanics had already been suggested, though in not very concrete terms, by Einstein, Podolsky and Rosen (EPR) \cite{epr}. These authors speculated that ``elements of physical reality'' with definite values exist irrespective of whether or not they are actually observed. Moreover, EPR conjectured, the quantum formalism can be ``completed'' or ``embedded'' into a larger theoretical framework which would reproduce the quantum theoretical results but would otherwise be classical and deterministic from an algebraic and logical point of view. A proper formalization of the term ``element of physical reality'' suggested by EPR can be given in terms of two-valued states or valuations, which can take on only one of two values $0$ and $1$ and which are interpretable as the classical logical truth assignments {\it false} and {\it true}, respectively. Recall that Kochen and Specker's results \cite{kochen1} state that for quantum systems representable by Hilbert spaces of dimension higher than two, there does not exist any such valuation $s: L\rightarrow \{0,1\}$ on the set of closed linear subspaces $L$ interpretable as quantum mechanical propositions preserving the lattice operations and the orthocomplement, even if these lattice operations are carried out among commuting (orthogonal) elements only. Moreover, the set of truth assignments on quantum logics is not separating and not unital. That is, there exist different quantum propositions which cannot be distinguished by any classical truth assignment. (For related arguments and conjectures based upon a theorem by Gleason \cite{Gleason}, see Zierler and Schlessinger \cite{ZirlSchl-65} and John Bell \cite{bell-66}.) Particular emphasis will be given to embeddings of quantum universes into classical ones which do not necessarily preserve (binary lattice) operations identifiable with the logical {\it or} and {\it and} operations. Stated pointedly, if one is willing to abandon the preservation of quite commonly used logical functions, then it is possible to give a classical meaning to quantum physical statements, thus giving rise to an ``understanding'' of quantum mechanics. One of the questions already raised in Specker's almost forgotten first article \cite{specker-60} concerned an embedding of a quantum logical structure $L$ of propositions into a classical universe represented by Boolean algebras $B$. Such an embedding can be formalized as a function $\varphi :L\rightarrow B$ with the following properties (Specker had a modified notion of embedding in mind; see below). Let $p,q\in L$. \begin{description} \item[(i)] Injectivity: two different quantum logical propositions are mapped into two different propositions of the Boolean algebra; i.e., if $p\neq q $ then $ \varphi (p)\neq \varphi (q)$. \item[(ii)] Preservation of the order relation: if $p\rightarrow q$ then $\varphi (p) \rightarrow \varphi (q)$. \item[(iii)] Preservation of the lattice operations, in particular preservation of the \begin{description} \item[(ortho-)complement] $\varphi(p')=\varphi (p)'$; \item[{\it or} operation] $\varphi (p\vee q)=\varphi (p) \vee \varphi (q)$; \item[{\it and} operation] $\varphi (p\wedge q)=\varphi (p) \wedge \varphi (q)$. \end{description} \end{description} It is rather obvious that we cannot have an embedding from the quantum to the classical universe satisfying all three requirements (i)--(iii). In particular, a head-on approach requiring (iii) is doomed to failure, since the nonpreservation of lattice operations among noncomeasurable propositions is quite evident, given the nondistributive structure of quantum logics. One method of embedding any arbitrary partially ordered set into a concrete orthomodular lattice which in turn can be embedded into a Boolean algebra has been used by Kalmbach \cite{kalmbach-77} and extended by Harding \cite{harding-91} and Mayet and Navara \cite{navara-95}. These {\it Kalmbach embeddings}, as they may be called, are based upon the following two theorems. Given any poset $P$, there is an orthomodular lattice $L$ and an embedding $\varphi :P\rightarrow L=K(P)$ such that if $x,y\in P$, then (i) $x\le y$ if and only if $\varphi (x)\le \varphi (y)$, (ii) if $x\wedge y$ exists, then $\varphi (x)\wedge \varphi (y)=\varphi (x \wedge y)$, and (iii) if $x\vee y$ exists, then $\varphi (x)\vee \varphi (y)=\varphi (x \vee y)$ \cite{kalmbach-77}.\footnote{Kalmbach's original result referred to an arbitrary lattice instead of the poset $P$, but by the MacNeille completion \cite{macneille} it is always possible to embedd a poset into a lattice, thereby preserving the order relation and the meets and joins, if they exist \cite{harding-priv}. Also, a direct proof has been given by Navara.} Furthermore, $L$ in the above result has a full set of two-valued states \cite{harding-91,navara-95} and thus can be embedded into a Boolean algebra $B$ by preserving lattice operations among orthogonal elements and additionally by preserving the orthocomplement. Note that the {\em combined} Kalmbach embedding $P\rightarrow K(P) \rightarrow B = P\rightarrow B$ does not necessarily preserve the logical {\it and}, {\it or} and {\it not} operations. (There may not even be a complement defined on the partially ordered set which is embedded.) Nevertheless, every chain of the original poset gets embedded into a Boolean algebra whose lattice operations are totally preserved under the combined Kalmbach embedding. The Kalmbach embedding of some bounded lattice $L$ into a concrete orthomodular lattice $K(L)$ may be thought of as the pasting of Boolean algebras corresponding to all maximal chains of $L$ \cite{harding-priv}. First, let us consider linear chains $0=a_0\rightarrow a_1\rightarrow a_2\rightarrow \cdots \rightarrow 1=a_m$. Such chains generate Boolean algebras $2^{m-1}$ in the following way: from the first nonzero element $a_1$ on to the greatest element $1$, form $A_n=a_n\wedge (a_{n-1})'$, where $(a_{n-1})'$ is the complement of $a_{n-1}$ relative to $1$; i.e., $(a_{n-1})'=1-a_{n-1}$. $A_n$ is then an atom of the Boolean algebra generated by the bounded chain $0=a_0\rightarrow a_1\rightarrow a_2\rightarrow \cdots \rightarrow 1$. Take, for example, a three-element chain $0= a_0\rightarrow \{a\}\equiv a_1\rightarrow \{a,b\}\equiv 1=a_2$ as depicted in Figure \ref{f-thech}a). In this case, \begin{eqnarray*} A_1&=&a_1\wedge (a_0)'=a_1\wedge 1\equiv \{a\}\wedge \{a,b\}=\{a\},\\ A_2&=&a_2\wedge (a_1)'=1\wedge (a_1)'\equiv \{a,b\}\wedge \{b\}=\{b\}. \end{eqnarray*} This construction results in a four-element Boolean Kalmbach lattice $K(L)=2^2$ with the two atoms $\{a\}$ and $\{b\}$ depicted in Figure \ref{f-thech}b). Take, as a second example, a four-element chain $0= a_0\rightarrow \{a\}\equiv a_1\rightarrow \{a,b\} \rightarrow \{a,b,c\}\equiv 1=a_3$ as depicted in Figure \ref{f-thech}c). In this case, \begin{eqnarray*} A_1&=&a_1\wedge (a_0)'=a_1\wedge 1\equiv \{a\}\wedge \{a,b,c\}=\{a\},\\ A_2&=&a_2\wedge (a_1)'\equiv \{a,b\}\wedge \{b,c\}=\{b\},\\ A_3&=&a_3\wedge (a_2)'=1\wedge (a_2)'\equiv \{a,b,c\}\wedge \{c\}=\{c\}.\\ \end{eqnarray*} This construction results in a eight-element Boolean Kalmbach lattice $K(L)=2^3$ with the three atoms $\{a\}$, $\{b\}$ and $\{c\}$ depicted in Figure \ref{f-thech}d). To apply Kalmbach's construction to any bounded lattice, all Boolean algebras generated by the maximal chains of the lattice are pasted together. An element common to two or more maximal chains must be common to the blocks they generate. Take, as a third example, the Boolean lattice $2^2$ drawn in Figure \ref{f-thech}e). $2^2$ contains two linear chains of length three which are pasted together horizontally at their smallest and biggest elements. The resulting Kalmbach lattice $K(2^2)=MO_2$ is of the ``Chinese lantern'' type, as depicted in Figure \ref{f-thech}f). Take, as a fourth example, the pentagon drawn in Figure \ref{f-thech}g). It contains two linear chains. One is of length three, the other is of length four. The resulting Boolean algebras $2^2$ and $2^3$ are again horizontally pasted together at their extremities $0,1$. The resulting Kalmbach lattice is depicted in Figure \ref{f-thech}h). In the fifth example drawn in Figure \ref{f-thech}i), the lattice has two maximal chains which share a common element. This element is common to the two Boolean algebras; and hence central in $K(L)$. The construction of the five atoms proceeds as follows. \begin{eqnarray*} A_1&=& \{a\}\wedge \{a,b,c,d\}=\{a\},\\ A_2&=& \{a,b,c\}\wedge \{b,c,d\}=\{b,c\},\\ A_3&=&B_3= \{a,b,c,d\}\wedge \{d\}=\{d\},\\ B_1&=& \{b\}\wedge \{a,b,c,d\}=\{b\},\\ B_2&=& \{a,b,c\}\wedge \{a,c,d\}=\{a,c\},\\ \end{eqnarray*} where the two sets of atoms $\{A_1,A_2,A_3=B_3\}$ and $\{B_1,B_2,B_3=A_3\}$ span two Boolean algebras $2^3$ pasted together at the extremities and at $A_3=B_3$ and $A_3'=B_3'$. The resulting lattice is $2\times MO_2=L_{12}$ depicted in Figure \ref{f-thech}j). Notice that there is an equivalence of the lattices $K(L)$ resulting from Kalmbach embeddings and automata partition logics \cite{svozil-93}. The Boolean subalgebras resulting from maximal chains in the Kalmbach embedding case correspond to the Boolean subalgebras from individual automaton experiments. 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\end{figure} \clearpage \ifx\undefined\bysame \newcommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\,} \fi \bibliography{svozil} \bibliographystyle{osa} %%\bibliographystyle{plain} %\bibliographystyle{aalpha} \end{document}