This is very different from the ``classical'' logical approach, which is also 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 [Mac57], Jauch [Jau68], Varadarajan [Var68,Var70], Piron [Pir76], Marlow [Mar78], Gudder [Gud79,Gud88], Maczy\'nski [Mac73], Beltrametti and Cassinelli [BC81], Kalmbach [Kal83,Kal86], Cohen [Coh89], Pták and Pulmannová [PP91], Giuntini [Giu91], and in a forthcoming book of the author [Svo98], among others. A bibliography on quantum logics and related structures has been compiled by Pavii\'c [Pav92].
``The physical system has a property corresponding to the associated closed linear subspace.''
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The symbol Å will used to indicate the closed linear subspace spanned by two vectors. That is,
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Let us verify some logical statements.
M(p¢)¢ = {x\mid (x,y) = 0, y Î {z\mid (z,u) = 0, u Î Mp } } = Mp.
M1¢ = {x\mid (x,y) = 0, y Î H} = M0 = 0.
M0¢ = {x\mid (x,y) = 0, y = 0} = M1 = H.
MpÚp¢ = Mp ÅMp¢ = {x \mid x = ay+bz, a,b Î C, y Î Mp, z Î Mp¢} = M1 .
MpÙp¢ = Mp ÇMp¢ = {x \mid x Î Mp, x Î Mp¢} = M0 .
| generic lattice | order relation ® | ``meet'' \sqcap | ``join'' \sqcup | ``complement'' ¢ |
| ``classical'' lattice | subset Ì | intersection Ç | union È | complement |
| of subsets | ||||
| of a set | ||||
| propositional | implication | disjunction | conjunction | negation |
| calculus | ® | ``and'' Ù | ``or'' Ú | ``not''Ø |
| Hilbert | subspace | intersection of | closure of | orthogonal |
| lattice | relation | subspaces Ç | linear | subspace |
| Ì | span Å | ^ | ||
| lattice of | E1E2 = E1 | E1E2 | E1+E2-E1E2 | orthogonal |
| commuting | (limn® ¥(E1E2)n) | projection | ||
| (noncommuting) | ||||
| projection | ||||
| operators | ||||
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 and and or operations among propositions. Points `` · '' 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 `` ° '' represent the atoms. 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.
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.
This construction is often referred to as pasting construction. If the blocks are only pasted together at the tautology and the absurdity, one calls the resulting logic a horizontal sum.2 In a sense, the pasting construction allows one to obtain a global representation of different universes which are defined (and classical) locally. This local - versus - global theme will be discussed below. A pasting of two propositional structures L1 and L2 will be denoted by
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The propositional system obtained is not a classical Boolean algebra, since the distributive law is not satisfied. This can be easily seen by the following evaluation. Assume that the distributive law is satisfied. Then,
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Notice that the expressions can be easily evaluated by using the Hasse diagram 1. For any a,b, aÚb is just the least element which is connected by a and b; aÙb 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,
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aÚb is called a least upper bound of a and b. aÙb is called a greatest lower bound of a and b.
MO2 is a specific example of an algebraic structure which is called a lattice. Any two elements of a lattice have a least upper and a greatest lower bound. Furthermore,
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It is an ortholattice or orthocomplemented lattice, since every element has a complement.
It is modular, since for all a® c, the modular law
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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 MOn of n classical Boolean algebras L(xi), where xi indicates the direction of a spin state measurement. That is,
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L(xi),i = 1,¼,n which are noncomeasurable. The superscript i represents the ith measurement direction.
L(xi),i = 1,¼,n which are noncomeasurable. The superscript i represents the ith measurement direction.
The finite subalgebras of two-dimensional Hilbert space are MOn,n Î N. This can be visualized easily, since given a vector v associated with a proposition pv, there exists only a single orthogonal vector v¢, corresponding to the proposition pv¢, which is the negation of the proposition pv. Conversely, the negation of pv¢ can be uniquely identified with the vector v.
This is not the case in three- and higherdimensional spaces, where the complement of a vector is a plane (or a higherdimensional 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 MOm [Kal]; e.g.,
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In particular, for n = 3, 21×MO2 = L12. In general, L2(2m+2) = L4m+4 = 21×MOm, and we are recovering the threedimensional case discussed before.
The logic 2n-2×MOm has a separating set of two-valued states. Therefore, it can be realized by automaton logics [Svo93].
The above class 2n-2×MOm, 1 < m Î N does not coincide with the class of all modular lattices corresponding to finite subalgebras of n-dimensional Hilbert logics for n > 3. Consider, for instance, four dimensional real Hilbert space R4. The product MO2×MO2 is a subalgebra of the corresponding Hibert logic but is not a logic represented by Equation (1). This can be demonstrated by identifying the following eight one-dimensional subspaces3
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This result could be generalized to the product MOi ×MOj by augmenting the above vectors with additional vectors (cosfl,sinfl,0,0), (sinfl,-cosfl,0,0), (0,0,cosfk,sinfk), (0,0,sinfk,-cosfk), such that all angles fl,k are mutually different, l,k Î N, and 1 < l £ i, 1 < l £ j.
Furthermore, the above considerations could be extended for evendimensional vector spaces by the proper multiplication of additional MO2 (MOm) factors. For instance, for six-dimensional Hilbert logic, we may consider three factors MO2 corresponding to
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Let us denote by C (Rn) the orthomodular lattice of all subspaces of Rn. This lattice is modular. Furthermore, any sublattice of C (Rn) is modular. As has been pointed out by Chevalier [Che],
B× Õi Î ILi where B is a Boolean algebra and the Li are simple (not isomorphic to a product) orthomodular lattices.
Rn = M1 Ú ... ÚMk and dim Mni = ni. Thus C(Rn1)× ... ×C (Rnk) is a sub-orthomodular lattice of C (Rn).
Let n > 0 be an integer. The finite sub-orthomodular posets of C (Rn) are the
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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 C(Rn).
Quantum logic suggests that the classical Boolean propositional structure of events should be replaced by the Hilbert lattice << of subspaces of a Hilbert space H. (Alternatively, we may use the set of all projection operators P (H).) 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 {Mpi } is any countable set of mutually orthogonal subspaces (corresponding to comeasurable propositions pi) having closed linear span MÚi pi = Åi Mpi, then
| (2) |
| (3) |
It is not difficult to show that a measure can be obtained by selecting an arbitrary normalized vector y Î H and by identifying Py(Mx) with the square of the absolute value of the scalar product of the orthogonal projection of y onto Mx spanned by the unit vector x,
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Indeed, a celebrated theorem by Gleason [Gle57] states that in a separable Hilbert space of dimension at least three, every probability measure on the projections satisfying (2) and (3) can be written in the form
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Then the expectation value of an observable corresponding to a self-adjoint operator A with eigenvalues li is (in n-dimensional Hilbert space) given by
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Gleason's theorem can be seen as a substitute for the probability axiom of quantum mechanics by deriving it from some ``fundamental'' assumptions and ``reasonable'' requirements. One such requirement is that, if Ep and Eq are orthogonal projectors representing comeasurable, independent propositions p and q, then their join pÚq (corresponding to Ep+Eq) has probability P(pÚq) = P(p)+P(q) (corresponding to P(Ep+Eq) = P(Ep)+P(Eq)).
In the late 50's, Ernst Specker was considering the question of whether it might be possible to consistently define elements of physical reality ``globally'' which can merely be measured ``locally'' [Spe60]. Specker mentions the scholastic speculation of the so-called ``infuturabilities''; that is, the question of whether the omniscience (comprehensive knowledge) of God extends to events which would have occurred if something had happened which did not happen (cf. [Spe60] and [Spe90]). 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 - Schrödinger's catalogue of expectation values [Sch35]. 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 6. 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.
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 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 context of measurement [Bel66,Red90] ; i.e., whether it is measured along with first or second frame of mutually compatible observables.5 It is in this sense that the nonexistence of two-valued probability measures is a formalization of the concept of context dependence or 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 R3 has been given by Peres [Per91]. (For other discussions, see Refs. [Sta83,Red90,Jam92,Bro92,Per91,Per93,ZP93,Cli93,Mer93,ST96].) 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
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The way it is defined, this set of lines is invariant under interchanges (permutations) of the x1,x2 and x3 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 7 [ST96]. For simplicity, 24 points which belong to exactly one edge are omitted. The coordinates should be read as follows: [`1]® -1 and 2® Ö2; e.g., 1[`1]2 denotes Sp (1,-1,Ö2). Concentric circles indicate the (non orthogonal) generators mentioned above.
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Let us prove that there is no two-valued probability measure [ST96,Tka96]. 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[`1]) = 1. It immediately follows that P(001) = P(010) = P(102) = P([`1]20) = 0. A second glance shows that P(20[`1]) = 1, P(1[`1]2) = P(112) = 0.
Let us now suppose (case 1a) that P(201) = 1. Then we obtain P([`1]12) = P([`1][`1]2) = 0. We are forced to accept P(110) = P(1[`1]0) = 1 - a contradiction, since (110) and (1[`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([`1]02) = 1 and P(211) = 0. Since P(01[`1]) = 0, we obtain P(2[`1][`1]) = 1 and thus P(120) = 0. This requires P(2[`1]0) = 1 and therefore P(12[`1]) = P(121) = 0. Observe that P(210) = 1, and thus P([`1]2[`1]) = P([`1]21) = 0. In the following step, we notice that P(10[`1]) = P(101) = 1 - a contradiction, since (101) and (10[`1]) are orthogonal to each other and lie on one edge.
Thus we are forced to assume (case 2) that P(2[`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 (6) orthogenerates (by the nor-operation between orthogonal vectors) a suborthoposet of R3 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 (±1,±1,Ö2) - plus their two-dimensional orthogonal planes plus the entire Hilbert space and the null vector [ST96].
This suborthoposet of R3 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,Ö2), (1,±1,Ö2). It has a 3-element set of generators
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[1] ` `= eqnarray* Sp(1,0,0)
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 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 commuting self-adjoint operators A and B corresponding to observables can be simultaneously diagonalized []. Furthermore, A and B commute if and only if there exists a self-adjoint ``Ur''-operator U and two real-valued functions f and g such that A = f(U) and B = g(U) (cf. [], Varadarajan [] and Pták and Pulmannová []). 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 = I and B = s1 (uncritical factors have been omitted). In this case, take U = B and f(x) = x2, 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 ( (h/2p) = 1)[]
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Let us be a little bit more explicit. We have
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We now consider then the following propositions
or equivalently,
For spin component measurements along a different set [`x],[`y],[`z] of mutually orthogonal rays, the ``Ur''-operator is given by
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Let us, for example, take [`x] = (1/Ö2)(1,1,0), [`y] = (1/Ö2)(-1,1,0), and [`z] = z. In terms of polar coordinates q,f,r, these orthogonal directions are [`x] = (p/2,p/4,1), [`y] = (p/2,-p/4,1), and [`z] = (0,0,1), and
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[`a]+[`b],[`b]+[`c],[`a]+[`c]. This result holds true for arbitrary rotations S0(3) [] of the coordinate axes (tripod), parameterized, for instance, by the Euler angles a,b, g.
Hence, stated pointedly and repeatedly, any measurement of elements of physical reality boils down, in a sense, to measuring a single ``Ur''-observable, from which the three observables in the tripod can be derived. Different tripods correspond to different ``Ur''-observables.
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) []. 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 false and true, respectively. Recall that Kochen and Specker's results [] state that for quantum systems representable by Hilbert spaces of dimension higher than two, there does not exist any such valuation s: L® {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 [], see Zierler and Schlessinger [] and John Bell [].)
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 or and 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 [] 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 j:L® B with the following properties (Specker had a modified notion of embedding in mind; see below). Let p,q Î L.
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 [] and extended by Harding [] and Mayet and Navara []. These 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 j:P® L = K(P) such that if x,y Î P, then (i) x £ y if and only if j(x) £ j(y), (ii) if xÙy exists, then j(x)Ùj(y) = j(x Ùy), and (iii) if xÚy exists, then j(x)Új(y) = j(x Úy) [].7 Furthermore, L in the above result has a full set of two-valued states [,] 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 combined Kalmbach embedding P®K(P) ® B = P® B does not necessarily preserve the logical and,
or and 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 [].
First, let us consider linear chains
0 = a0®a1® a2® ¼® 1 = am. Such chains generate Boolean algebras 2m-1 in the following way: from the first nonzero element a1 on to the greatest element 1, form An = anÙ(an-1)¢, where (an-1)¢ is the complement of an-1 relative to 1; i.e., (an-1)¢ = 1-an-1. An is then an atom of the Boolean algebra generated by the bounded chain
0 = a0®a1® a2® ¼® 1.
Take, for example, a three-element chain 0 = a0® {a} º a1®{a,b} º 1 = a2 as depicted in Figure 8a). In this case,
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Take, as a second example, a four-element chain 0 = a0® {a} º a1® {a,b}® {a,b,c} º 1 = a3 as depicted in Figure 8c). In this case,
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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 22 drawn in Figure 8e). 22 contains two linear chains of length three which are pasted together horizontally at their smallest and biggest elements. The resulting Kalmbach lattice K(22) = MO2 is of the ``Chinese lantern'' type, as depicted in Figure 8f).
Take, as a fourth example, the pentagon drawn in Figure 8g). It contains two linear chains. One is of length three, the other is of length four. The resulting Boolean algebras 22 and 23 are again horizontally pasted together at their extremities 0,1. The resulting Kalmbach lattice is depicted in Figure 8h).
In the fifth example drawn in Figure 8i), 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.
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Notice that there is an equivalence of the lattices K(L) resulting from Kalmbach embeddings and automata partition logics []. The Boolean subalgebras resulting from maximal chains in the Kalmbach embedding case correspond to the Boolean subalgebras from individual automaton experiments. In both cases, these blocks are pasted together similarly.