aa r X i v : . [ qu a n t - ph ] A ug AN AXIOMATIC BASIS FOR QUANTUM MECHANICS
GIANNI CASSINELLI AND PEKKA LAHTI
Abstract.
In this paper we use the framework of generalizedprobabilistic theories to present two sets of basic assumptions,called axioms, for which we show that they lead to the Hilbertspace formulation of quantum mechanics. The key results in thisderivation are the co-ordinatization of generalized geometries anda theorem of Solér which characterizes Hilbert spaces among theorthomodular spaces. A generalized Wigner theorem is applied toreduce some of the assumptions of Solér’s theorem to the theoryof symmetry in quantum mechanics. Since this reduction is onlypartial we also point out the remaining open questions.PACS numbers: 03.65.-w Introduction
This paper aims to give an overview of the axiomatic basis of quan-tum mechanics. We show that quantum mechanics on Hilbert space canto a large extent be derived from physically motivated assumptions us-ing either the quantum logic approach or the convexity approach, bothbeing examples of general probabilistic theories. The key results inthese derivations are the coordinatization of generalized geometries, atheorem of Solér, and a generalized Wigner theorem. We also point outa mathematical assumption, which seems unavoidable but still lacks anoperational justification.The historic paper [4] of Birkhoff and von Neumann, entitled
Thelogic of quantum mechanics , marks the beginning of the investigationson the mathematical and conceptual foundations of quantum mechan-ics which go under the title quantum logic. The literature of the fieldis very rich. In addition to the influential lecture notes of Mackey [43]we mention here only some representative monographs to indicate thediversity of the field [3, 30, 33, 48, 51, 56, 57, 64].The papers of Ludwig [41], Mielnik [46, 47], Davies and Lewis [15],and Edwards [18, 19, 20] have strongly influenced the developmentof the convexity or operational approach to quantum mechanics. Inaddition to the mentioned original papers, the monumental work ofLudwig [42] as well as the monographs [31, 16] are valuable sources forthe physical and mathematical ideas behind the convextity approaches. closely related approach is the empirical logic framework developedby Foulis and Randall to study the manuals of physical operations; see,for instance, their papers [25, 26].In recent years quantum information theory has renewed interestin the foundations of quantum mechanics and some new ideas havebeen proposed for an axiomatic foundations of quantum mechanics,see, e.g., [11, 12, 13] and the many references given therein. Thoughinteresting, these investigations deal only with the so-called finite levelsystems. Finite level systems are only a part of quantum mechanics,leaving open the more general question on the derivability of quantummechanics in infinite dimensional Hilbert space, which is needed, forinstance, if one assumes that physical systems exist in four dimensionalspacetime R . Our aim is to investigate this problem.The structure of the paper is the following. The first part of the pa-per, Sections 2 - 4, discusses a general probabilistic framework for anaxiomatic basis of quantum mechanics. Section 2 reviews the idea ofstatistical duality, the concepts of states, experimental functions, op-erations, and effects, and defines the basic structures. We summarizethem in two pairs of axioms, the weaker ones 1 and 3, and the strongerones 2 and 4, concerning the sets of states S and experimental func-tions E . In Section 3 quantum logic is defined as a pair ( S , L ) , L ⊂ E ,arising from the structure ( S , E ) of Axioms 1 and 3 as specified furtherthrough the orthogonality postulate (Axioms 5) and an axiom whichstipulates the existence of a sufficiently rich family P of pure states,extremal elements in S , expressed as probability measures on L (Ax-iom 6). After excluding the classical case in Section 3.3, Corollary 1concludes that the fundamental structure attached to a proper quan-tum system consists of L being an irreducible complete orthomodularAC lattice where the atoms of L are in a bijective correspondence withthe pure states in S . As an alternative approach, Section 4 starts withthe stronger pair of axioms, 2 and 4, defining a subset of operations,called filters, together with a subset of experimental functions, calledpropositions, and poses the projection postulate, Axiom 7, to build aone-to-one onto connections between the sets of filters and propositions.This leads, once again, to the structure of Corollary 1.The two final Sections 5 - 6 constitute the second part of the paper.There we study the problem of realizing the abstract structure ( S , E ) ,with the substructure ( P , L ) of Corollary 1, as the one given by theHilbert space formulation of quantum mechanics. In this realization, E is identified with the set of effect operators (positive unit boundedoperators), L as the extremal elements of E , that is, the projectionoperators, S , via Gleason’s theorem, as the density operators (positive race one operators), and P as the extremal elements of S , the one-dimensional projections, with tr (cid:2) ρE (cid:3) giving the probability for an effect E in state ρ .To obtain the Hilbert space realization of ( S , E ) we follow the some-what indirect, and presumably not optimal, method that starts byidentifying L with the lattice of the closed subspaces of an orthomod-ular space over a division ring, Sect. 5.2. Exhibiting an example of afinite level non-Hilbertian model with the structure of Corollary 1, weproceed to apply a theorem of Solér (Sect. 5.3) to fix the orthomodu-lar space to be an infinite dimensional (real, complex, or quaternionic)Hilbert space. The final Section 6 reduces the assumptions required bySolér’s theorem to the idea of symmetry, indicating, at the same timean unavoidable assumption whose physical meaning remains yet to beclarified. The choice between real, complex and quaternionic cases isbriefly discussed at the end of the paper.2. Statistical duality and its representation
Statistical duality.
A general probabilistic formulation of a phys-ical theory builds on the concepts of states and observables and on theidea of statistical causality merging the two entities into a probabilitymeasure.
States are understood as equivalence classes of preparationsof a physical system, observables as equivalence classes of measure-ments on it, and the statistical causality claims that any state α andany observable E determine a probability measure p ( α, E , · ) defined ona σ -algebra A of subsets of a (nonempty) set Ω , with p ( α, E , X ) givingthe probability that a result is registered in the set X when a mea-surement of E is performed on the system prepared in state α . Anobservable E thus goes with a value space Ω together with the test sets X ∈ A within which the results are counted. To emphasize this, wemay also write ( E , Ω , A ) for E . In most applications (Ω , A ) is just thereal Borel space ( R , B ( R )) , or a Cartesian product space ( R n , B ( R n )) ,or a (Borel) subspace of such spaces.Let S and O be the sets of all states and all observables of thesystem. We call the pair ( S , O ) together with the probability function p : ( α, E ) p ( α, E , · ) a statistical duality . In an axiomatic approachone aims to introduce physically motivated structures for the sets S and O so that the form of the probability measures p ( α, E , · ) , α ∈ S , E ∈ O ,gets determined.2.2. State space. .2.1. Convex structure.
The set S of states can immediately be equippedwith a convex structure reflecting the possibility of combining prepa-rations (and thus states) into new preparations (and thus states) bystatistically mixing them. Indeed, for any α, β ∈ S and ≤ λ ≤ onemay define a state h λ, α, β i through the λ -convex combination of theprobability measures p ( α, E , · ) and p ( β, E , · ) , E ∈ O ,(1) p ( h λ, α, β i , E , · ) = λp ( α, E , · ) + (1 − λ ) p ( β, E , · ) . The existence of such a state h λ, α, β i is built in the assumption thatpreparations can be statistically mixed to produce new preparationsand the uniqueness of h λ, α, β i follows from the statistical completenessbuilt in the notions of states (as equivalence classes of preparations)and observables (as equivalence classes of measurements).The existence of the function [0 , × S × S → S with the property(1) defines what is known as a convex structure , and it is then a math-ematical convenience to consider S as properly placed in a real vectorspace U so that we may simply write h λ, α, β i = λα + (1 − λ ) β [63,Theorem 2], see also [29, 14].Using σ -convex combinations of probability measures one may alsointroduce σ -convex combinations of states ( α i ) with weights ( λ i ) , λ i ≥ , P λ i = 1 , through P λ i p ( α i , E , · ) , E ∈ O , with the obvious require-ment that the series P λ i p ( α i , E , X ) is convergent for each E , X . Ifthere is an α ∈ S such that p ( α, E , · ) = P λ i p ( α i , E , · ) for all E ∈ O , wesay that α is a σ -convex combination of the states ( α i ) with weights ( λ i ) and we write α = P λ i α i . Again, if such a state exists it is unique.As seen from Theorem 1 below it is a mathematically convenient ide-alization to assume that the set of states is also closed under σ -convexcombinations.The convex structure of S allows the distinction between the purestates , the extreme elements of S , and the mixed states , the nonextremeelements of S . We let P = ex( S ) denote the set of all pure states in S . The existence of pure preparations and thus pure states is anothernatural assumption supported equally well by everyday experience aswell as by sophisticated quantum experiments.We summarize the above discussion in the first axiom. Axiom 1.
The set of all states of a physical system described by thestatistical duality ( S , O , p ) forms a convex subset of a real vector space. We specify later the assumptions concerning σ -convex combinationsof states as well as the existence of a sufficiently large set of pure states.2.2.2. S as a base for a generating cone. Let K = { λα | λ ∈ R + , α ∈ S } = ∪ λ ≥ λ S ⊂ U be the cone defined by S . We assume now that this one is a proper cone, that is, K ∩ − K = { } , and that each γ ∈ K , γ = 0 , has a unique representation as γ = λα for some λ > and α ∈ S . Let V = K − K be the real vector space generated by K (possibly a subspace of U ). Then K = V + = { v ∈ V | v ≥ } and S is a base of the cone K . The existence of a base for a generating coneof a real vector space V is known to be equivalent with the existenceof a strictly positive linear functional on V [22, Lemma 2]. We let e : V → R be the functional defined by S so that(2) S = { α ∈ V + | e ( α ) = 1 } and call it the intensity functional.The physical interpretation attached to S can be extended to thepositive cone V + : any λα , λ ∈ R + , α ∈ S , represents a new state ofthe system obtained from α by changing its intensity. The elements of S will be distinguished as normalized states, whereas the term ‘state’will be extended to refer to all elements of V + , including the emptystate, the null element of V . The linear operations ( α, β ) α + β and ( λ, α ) λα , α, β ∈ V + , λ ∈ R + , preserve their original interpretationas mixing and intensity changing, respectively. In particular, the term pure state can thus refer to an element of ex( S ) or of ed( V + ) = { λα | λ ∈ R + , α ∈ ex( S ) } , where ed stands for ‘edge’. This extension will beaccepted only as a mathematically convenient way of speaking and ithas no physical implications.Let conv( S ∪ − S ) denote the convex hull of the set S ∪ − S . Thisset is convex, absorbing (for any v ∈ V there is a λ > such that v ∈ λ conv( S ∪ − S ) ), and balanced ( λv ∈ conv( S ∪ − S ) for all v ∈ conv( S ∪ − S ) and − ≤ λ ≤ ). Therefore its Minkowski functional(or gauge) p S : V → R , defined as p S ( v ) = inf { λ > | v ∈ λ conv( S ∪ − S ) } , is a seminorm [58, Theorem II.1.4]. Clearly, p S ( α ) = e ( α ) for all α ∈ V + and we note that for all v ∈ Vp S ( v ) = inf { e ( α ) + e ( β ) | α, β ∈ V + , v = α − β } . If this seminorm is a norm, then ( V, V + , S ) or just ( V, S ) is a base normspace . The following result, due to Edwards and Gerzon [21] is nowcrucial: Theorem 1.
If the set S of all states of the system forms a base for agenerating cone of a vector space V and is σ -convex, then its Minkowskifunctional p S : V → R is a norm with respect to which V is a Banachspace. here is another technical point to be noted. According to a proposi-tion of Ellis [23], if ( V, V + , S ) is a base norm space, then also ( V, V + , S ) (where A denotes the norm closure of a subset A of V ) is a base normspace and its norm coincides with the norm of ( V, S ) . We note, inaddition, that S is closed if and only if V + is closed. With the riskof adding some nonphysical elements in the set S we now formulatean alternative stronger assumption concerning the set of states of astatistical duality, as first formulated in [15]. Axiom 2.
The set of all states of a physical system described by thestatistical duality ( S , O , p ) is represented by a norm closed generatingcone V + of a base norm Banach space ( V, S ) . We have formulated two different axioms concerning the basic as-sumptions on the set of states, the weaker axiom serving as a startingpoint for the quantum logic approach, Section 3, the stronger axiomdefining the beginning of the convexity or state space approach, Section4.2.3.
Experimental functions.
Affine maps S → [0 , . With every observable ( E , Ω , A ) ∈ O one may consider the family of all ordered pairs h E , X i , X ∈ A , calledexperimental pairs. With each such pair we can associate the statement‘a measurement of E yields a result in the set X ’, denoted by ( E , X ) andcalled an experimental statement . Then the number p ( α, E , X ) , α ∈ S gives the probability for the statement ( E , X ) to be true in the state α .Experimental statements ( E , X ) and ( F , Y ) are said to be equivalent iffor all α ∈ S ,(3) p ( α, E , X ) = p ( α, F , Y ) . This defines an equivalence relation in the set of all experimental state-ments { ( E , X ) | E = ( E , Ω , A ) ∈ O , X ∈ A} . Let E denote the set ofall equivalence classes | ( E , X ) | of the statements ( E , X ) . A given el-ement of E is denoted by a letter a and is called an (experimental)proposition . The experimental proposition a is a set of experimentalstatements equivalent among themselves with respect to p .There is a fundamental difference between ( E , X ) and | ( E , X ) | . Namely, ( E , X ) is nothing more than a statement saying that a measurementof E yields a result in X , and it does not depend on p ; in contrast, | ( E , X ) | represents the proposition that every two statements ( E , X ) and ( E , X ) from | ( E , X ) | are equivalent to ( E , X ) , that is, for every α ∈ S , p ( α, E , X ) = p ( α, E , X ) = p ( α, E , X ) . The experimentalproposition | ( E , X ) | clearly depends on p ; it should be written more xactly | ( E , X ) | p . Hence to specify the proposition | ( E , X ) | p one has tomeasure of all the experimental statements ( F , Y ) and find those equiv-alent to ( E , X ) . Hence the experimental proposition a = | ( E , X ) | p is asubset of all experimental statements depending on p , typically muchbigger than the one-element set { ( E , X ) } .Each experimental proposition a ∈ E defines a real valued function, experimental function (4) f a : S → [0 , , f a ( α ) = p ( α, E , X ) , ( E , X ) ∈ a, and we let E ⊂ [0 , S denote the set of all such functions. If f ∈ E ,then f ( α ) , α ∈ S , is the probability that the proposition a = | ( E , X ) | ,with f = f a , is true in state α , that is, an E -measurement in state α yields a result in X .Consistently with the very definition (1) of mixed states it is naturalto assume that the experimental functions are affine, that is, we havethe following axiom. Axiom 3.
The set E of experimental functions of a statistical duality ( S , O , p ) is a subset of the set of affine functions S → [0 , . We let and e denote the constant zero and one functions S → [0 , ,respectively. Clearly, , e ∈ E , and if f ∈ E then also f ⊥ = e − f ∈ E so that for any α ∈ S , f ( α ) + f ⊥ ( α ) = 1 . Moreover, as real valuedfunctions, the set E is partially ordered in a natural way, that is, forany f, g ∈ E , f ≤ g if and only if f ( α ) ≤ g ( α ) for all α ∈ S , theoperational content being given by (4).The set E has the order bounds 0 and e and the mapping E ∋ f f ⊥ ∈ E is an order reversing involution. However, it is notan orthocomplementation, that is, the greatest lower bound of a pair ( f, f ⊥ ) need not be 0.Any observable E can now be represented as an E -valued set func-tion E : X f | ( E ,X ) | such that for each α ∈ S , the set function X α ( f | ( E ,X ) | ) = f | ( E ,X ) | ( α ) = p ( α, E , X ) is a probability measure.Moreover, any f ∈ E is in the range of some observable E . In thissense the set O of all observables is a surjective set of E -valued setfunctions.We shall follow two distinct approaches to specify further the struc-ture on E . For that end, we already pose the following two definitions: Definition 1.
A sequence (finite or countably infinite) of experimentalfunctions f , f , . . . is orthogonal if there is an experimental function g such that g + f + f + . . . = e . efinition 2. A sequence (finite or countably infinite) of experimentalfunctions f , f , . . . is pairwise orthogonal if f i + f j ≤ e for all i = j , i, j = 1 , , . . . An orthogonal sequence is pairwise orthogonal but in general notconversely. Still the concepts of orthogonality and pairwise orthogo-nality have the following common property: if for some state α one ofthe experimental statements f i is true, that is, f i ( α ) = 1 , then all theother experimental statements are false in that state, that is, f j ( α ) = 0 for all j = i . This supports the hypothesis that the statistical duality ( S , O , p ) might have a substructure where the two notions coincide; wereturn to that in Sect. 3.2.3.2. Positive unit bounded functionals on V . Further properties ofexperimental functions can be obtained under the assumption of Ax-iom 2. Indeed, in this case any f ∈ E has a unique extension to apositive continuous linear functional on V bounded by e . We denotethis extension with the same symbol f . In this case, the set E of ex-perimental functions is thus a subset of the order interval [0 , e ] of theorder unit Banach space ( V ∗ , [0 , e ]) . With the risk of adding some newelements in the set O of all observables one could assume that actu-ally E = [0 , e ] . We express also this stronger assumption concerningexperimental functions as a further possible axiom.
Axiom 4.
The set E of all experimental functions coincides with theorder interval [0 , e ] of the dual Banach space of ( V, S ) . This axiom has a simple but important consequence: for any two f, g ∈ E , if f ≤ g ⊥ , then also f + g ∈ E . Clearly, then f, g ≤ f + g ,but this does not mean that their smallest upper bound f ∨ g wouldexist in E , and even if it would exist, it need not equal to f + g .Another important structure of the set E arising from Axiom 4 isits convexity; for any f, g ∈ E and ≤ λ ≤ , λf + (1 − λ ) g ∈ E . Since the order interval E ⊂ V ∗ is also compact (by the Banach-Alaoglu theorem), the Krein-Milman theorem says that the closureof the convex hull of the extremal elements of E is the whole set ofexperimental functions, that is, conv(ex( E )) = E .2.4. Operations.
The number p ( α, E , X ) = f | ( E ,X ) | ( α ) is the probabil-ity that a measurement of E ∈ O in the state α ∈ S leads to a result inthe set X ∈ A . Such a measurement may destroy the system or, in anycase, cause a change in its state. In addition to such a forced change Clearly, this assumption could also be posed under Axiom 1 but we refrain ofdoing it. he system may also experience a spontaneous change, for instance, inthe course of its time evolution.To describe such state changes we now build on axioms 2 and 4. It isalso convenient to allow the possibility that the intensity of a state maychange in the process: V + ∋ α α ′ ∈ V + , including the possibilitythat α ′ = 0 , that is, the system gets destroyed in the intervention. Weconsider only such changes on the system which can be described byfunctions V + ∋ α φ ( α ) ∈ V + , with the obvious interpretation that α is the state of the system before the change and φ ( α ) its state afterthe change. Various types of state changes α φ ( α ) may occur inrealistic physical situations.We restrict our consideration only to such state changes where theintensity of the state is not increasing, that is, we assume that for eachstate α ∈ V + ,(5) e ( φ ( α )) ≤ e ( α ) . Consider then a mixed state β = λ α + λ α , with λ , λ ∈ R + and α , α ∈ V + . In a change φ the state β transforms to φ ( β ) whereasthe states α , α transform to φ ( α ) and φ ( α ) of which one may formthe mixture λ φ ( α ) + λ φ ( α ) . There are physical situations wherethe state φ ( λ α + λ α ) may differ from the state λ φ ( α ) + λ φ ( α ) . Again, we restrict our attention only to such changes φ for which thesestates are always the same, that is, for any λ , λ ∈ R + , α , α ∈ V + ,(6) φ ( λ α + λ α ) = λ φ ( α ) + λ φ ( α ) . It is again a simple exercise to check that any map φ : V + → V + withthe properties (5) and (6) has a unique extension to a positive linearcontracting mapping of V into V . We denote this extension by thesame letter φ and we call such mappings operations .Let O be the set of all operations. The sequential application ofany two operations defines a new operation giving O the structure of anoncommutative semigroup. Another physically relevant structure of O is that of convexity, for any two φ , φ ∈ O and for any ≤ λ ≤ , λφ + (1 − λ ) φ ∈ O , which allows one to single out the extremaloperations, that is, operations that cannot be obtained as nontrivialmixings of any other operations.An operation φ ∈ O , when combined with the intensity functional e , defines an experimental function e ◦ φ ∈ E . On the other hand, if f ∈ E , then fixing a β ∈ S and defining φ ( α ) = f ( α ) β , α ∈ V , oneobserves that φ ∈ O and e ◦ φ = f . The set of functionals e ◦ φ, φ ∈ O , The paper of Mielnik [46] contains an extensive analysis of possible statechanges, including some nonlinear processes. hus coincides with the set of experimental functions. Due to thiscoincidence, the experimental functions are also called effects : f ∈ E is the effect of any operation φ ∈ O such that f = e ◦ φ . Extremalelements of E are called decision effects or sharp effects; hence, an effectis either sharp (extremal) or unsharp (non-extremal).Let f be a decision effect. If f = e ◦ ( λφ + (1 − λ ) φ ) , for some φ , φ ∈ O , λ ∈ [0 , , then e ◦ φ = e ◦ φ , that is, the operations φ and φ are isotonic . Actually, the relation e ◦ φ = e ◦ ψ , φ, ψ ∈ O , defines anequivalence relation in O and one may immediately confirm that thereis a one-to-one onto correspondence between the set of effects and theisotony classes of operations. We let [ φ ] f denote the isotony class ofoperations φ ∈ O associated with the effect f ∈ E .One may now define an instrument as an operation valued set func-tion A ∋ X φ X ∈ O for which X e ◦ φ X is an observable, that is, X e ( φ X ( α )) is a probability measure for each α ∈ S . By definition,any instrument defines an observable, but the converse holds also: anyobservable E arises from some instrument such that E ( X ) = e ◦ φ X .Calling two instruments isotonic if they define the same observable oneagain has that the isotony classes of instruments are in one-to-one ontocorrespondence with the observables of the system.Preparing the system in a state α ∈ S , acting on it by an operation φ ∈ O , and detecting the (probabilistic) effect e ◦ φ ∈ E comprisesthe main steps in the operational approach built on the statistical du-ality of states and observables. To specify further stuctures of thedescription ( S , O , E ) one may proceed in many different ways by pos-ing fadditional conditions on any of the sets S , O , or E . The remarkbelow is an indication how to reach classical descriptions out of thisgeneral probabilistic model. Our aim is to pose conditions which leadto quantum descriptions. Remark 1.
The space V is ordered by the cone V + defined by the base S . If this order is a lattice order, then S is a (Choquet) simplex [1] ,a structure considered to be characteristic of classical descriptions. If V = V + − V + is a vector lattice, then also its dual V ∗ is a vectorlattice. In this case also the order interval E is a lattice and the set ex( E ) of extremal effects is a Boolean lattice with f f ⊥ = e − f as the orthocomplementation [59] - another characteristic of classicaldescriptions. In the next two chapters we shall follow two different approachesto specify further the statistical duality ( S , O , p ) . We start with anapproach based on axioms 1 and 3. In Section 4 we build on thestronger axioms 2 and 4. . Quantum logic
The Mackey approach to quantum logic can be viewed as a fur-ther specification of the structures arising from the statistical duality ( S , O , p ) with assuming, in the first instance, the existence of a suffi-ciently large subset of observables e O ⊆ O for which the order structureof the resulting subset of experimental functions gets sufficiently regu-lar. In this section we discuss assumptions of this kind. We stress oncemore that here we assume only that the set of states is convex and weallow the possibility that the set of experimental functions is a subsetof the set of affine functions S → [0 , .3.1. Orthogonality postulate.
Any subset of the set O of all ob-servables defines the corresponding subsets of the sets E and E . Thebasic assumption of the Mackey approach to quantum logic (Mackey’s[43] Axiom V) can now be restated as the requirement on the exis-tence of a (nonempty) subset e O ⊆ O such that in the resulting subset L ⊆ E of the experimental functions the two notions of Definitions 1and 2 coincide. We call this assumption the orthogonality postulate andformulate it as a further axiom. Axiom 5.
The set O of observables of the statistical duality ( S , O , p ) contains a (nonempty) subset e O such that in the resulting set L ofexperimental functions a sequence f , f , . . . ∈ L is orthogonal (in L ) ifand only if it is pairwise orthogonal (in L ).This axiom has important implications in the order structure of theset L . First of all, it guarantees that for any two mutually orthogonalelements f, g ∈ L also f + g ∈ L . Moreover, it implies that the map f f ⊥ is an orthocomplementation and it turns ( L , ≤ , ⊥ ) into anorthomodular σ -orthocomplemented partially ordered set, with , e ∈ L as the order bounds.Though obvious, we note that for any two f, g ∈ L , the set of theirlower (upper) bounds in L is smaller than the corresponding set in E .Therefore, f ∧ g may exists in L without existing in E . Theorem 2. (M¸aczynski, [44] ) Let ( S , O , p ) be a statistical duality andlet e O be a (nonempty) subset of O such that the associated set L ofexperimental functions satisfies Axiom 5. The set L is an orthocom-plemented orthomodular σ -orthocomplete partially ordered set with re-spect to the natural order of real functions and the complementation f ⊥ = e − f . roof. Clearly, L is partially ordered by ≤ and , e ∈ L . With f ∈ L also f ⊥ = e − f ∈ L , and L ∋ f f ⊥ ∈ L is an order reversinginvolution.Let f , f ∈ L and assume that f + f ≤ e . Then by Axiom 5 f = f + f ∈ L . To show that f is the least upper bound of f and f in L , assume that g ∈ L is such that f ≤ g and f ≤ g .Then also f + g ⊥ ≤ e and f + g ⊥ ≤ e and thus f + f + g ⊥ ∈ L so that f + f ≤ g , that is f + f = f ∨ L f . By induction oneshows that f + · · · + f n = f ∨ L · · · ∨ L f n for any pairwise orthogonalset { f , . . . , f n } ⊂ L . Let ( f i ) be a sequence of mutually orthogonalelements in L so that by assumption f = f + f + . . . ∈ L . Clearly, f i ≤ f for each i . Let g ∈ L be such that f i ≤ g for all i . Since for any n , f + · · · + f n = f ∨ L · · · ∨ L f n we thus have f + · · · + f n ≤ g for any n = 1 , , . . . and therefore f = f + f + . . . ≤ g .For any f ∈ L , f + f ⊥ ≤ e and thus e = f + f ⊥ = f ∨ L f ⊥ . By deMorgan laws we also have f ∧ L f ⊥ = 0 for any f ∈ L . This concludesthe proof that L is orthocomplemented and σ -orthocomplete.To show orthomodularity, we need to show that for f ≤ g , f, g ∈ L ,one has g = f ∨ L ( g ∧ L f ⊥ ) . If f ≤ g , then f ∨ L g ⊥ = f + g ⊥ = f +( e − g ) and h = ( f ∨ L g ⊥ ) ⊥ = g − f ∈ L . Hence f + h = g ≤ e and thus f ∨ L h = f + h so that f ∨ L ( g ∧ L f ⊥ ) = f ∨ L ( f ∨ L g ⊥ ) ⊥ = f + h = g . (cid:3) We call L the logic of p . Henceforth we simply write f ∧ g for f, g ∈ L instead of f ∧ L g , and similarly for f ∨ g , whenever the meet (join) existsin L . Remark 2.
Consider an f ∈ L , = f = e , and assume that λf ∈ L for some < λ < . Since λf ≤ f , then λf + ( e − f ) ∈ L , and hence e − ( λf + ( e − f )) = (1 − λ ) f ∈ L . Since λf + (1 − λ ) f = f ≤ e , also λf and (1 − λ ) f are pairwise orthogonal so that their sum should equal totheir least upper bound in L , which is a contradiction. In particular, L is not convex.Each observable E = ( E , Ω , A ) ∈ e O determines a unique L -valuedmeasure M E : A → L defined by M E ( X ) = f | ( E ,X ) | . By Theorem 2, M E is in fact a σ -homomorphism implying, in particular, that M E ( A ) is aBoolean sub- σ -algebra of L . We identify M E with E .Each state α ∈ S determines a unique probability measure m α : L → [0 , defined by m α ( f ) = f ( α ) , meaning, in particular, that forany pairwise orthogonal sequence ( f i ) in L , m α ( ∨ i f i ) = P i m α ( f i ) .Again, we identify m α with α .The family of L -valued measures M E , E ∈ e O , is surjective (that is,any f ∈ L is of the form f = M E ( X ) for some M E ( X ) ), and the family f probability measures m α , α ∈ S , is order determining , that is, forany f, g ∈ L , f ≤ g if and only if m α ( f ) ≤ m α ( g ) for all α ∈ S . Foreach α ∈ S , E ∈ e O , X ∈ A we have p ( α, E , X ) = m α ( M E ( X )) . We note that also the converse result is true: If L is an arbitraryorthocomplemented partially ordered set admitting an order determin-ing set of probability measure S , and e O is a surjective set of L -valuedmeasures, then the function p defined as p ( α, M , X ) = α ( M ( X )) forall α ∈ S , M ∈ e O , X ∈ A , is a probability function satisfying theorthogonality postulate and the logic of p is isomorphic to L [44].The sets L and L of experimental propositions and functions arein one-to-one onto correspondence and one may immediately trans-form the order and complementation of L to L : for any a, b ∈ L , a ≤ b if and only if f a ≤ f b , and we let a ⊥ stand for the propositioncorresponding to the function e − f a . Thus, under the assumption ofAxiom 5, we may equally well consider L as an orthocomplemented σ -orthocomplete orthomodular partially ordered set (of propositions),with S as an order determining set of probability measures of L . Fromnow on we do not distinguish between L and L and we also inter-changeably consider the elements of L as functions on S and the el-ements of S as functions on L : a ( α ) = α ( a ) . Also, together with L we always mean the structure ( L , ≤ , ⊥ ) , with the order bounds and e , corresponding to the absurd (always false) and trivial (always true)propositions. Moreover, we view the observables ( E , Ω , A ) ∈ e O as L -valued measures and we recall that for each a ∈ L there is an observable E ∈ e O and a set X ∈ A such that a = E ( X ) .An important technical assumption concerning the structure of L isthe separability of L ; this is the property that any pairwise orthogonalsequence ( a i ) ∈ L is at most countably infinite. This structure hasthe following measurement theoretical justification. The range E ( A ) ofany observable E ∈ e O is a Boolean sub- σ -algebra of L . If the valuespace (Ω , A ) of E ∈ e O is a subspace of the real Borel space ( R n , B ( R n )) ,for some n = 1 , , . . . , then the Boolean σ -algebra E ( A ) is separable.By the classic Loomis-Sikorski theorem, any separable Boolean sub- σ -algebra B ⊂ L is the range of some (real valued) observable E : B ( R ) → L [64]. If the logic L is separable then any Boolean sub- σ -algebra of L is also separable and thus appears as the range of an observable. Withthis motivation we pose the following assumption: Separability of the logic : Any orthogonal sequence ( a i ) ⊂ L isat most countably infinite. e call the pair ( S , L ) the logic of the statistical duality ( S , O , p ) associated with a subset e O of observables satisfying the orthogonalitypostulate, Axiom 5. We also assume that the logic is separable.3.2. Further specifications.
The set S of states is convex and itdetermines the order on L . We now assume that pure states exist.Moreover, we assume that there are sufficiently many so that each a ∈ L , a = 0 , can be realized in some pure state, that is, there is an α ∈ P such that α ( a ) = 1 . Sufficiency of pure states : For any a ∈ L , a = 0 , there is an α ∈ P such that α ( a ) = 1 .The Jauch-Piron property is a further important property of thelogic ( S , L ) : The Jauch-Piron property : For any a, b ∈ L , if α ( a ) = α ( b ) = 1 for some α ∈ S , then there exists a c ∈ L such that c ≤ a , c ≤ b and α ( c ) = 1 .These two assumptions have strong structural implications. To statethe relevant result we recall that an element a ∈ L is the support ofthe state α ∈ S if α ( a ) = 1 and for any b ∈ L the condition α ( b ) = 1 implies b ≥ a , that is, a , if exists, is the smallest proposition whichis true (in the sense of probabilistic certainty) in the state α . If thesupport of α exists it is unique and we donote it by s ( α ) . Theorem 3.
If the set S of states of the logic ( S , L ) contains a suffi-cient set P of pure states and satisfies the Jauch-Piron property then L is a complete orthocomplemented orthomodular lattice. Each state α ∈ S has a support s ( α ) ∈ L and each a ∈ L , a = 0 , is a support ofsome state α ∈ S . Proof. We show first that each α ∈ S has a support in L . If α ( a ) = 0 for any a ∈ L , then α ( a ) < for each a = 1 , meaning that s ( α ) = 1 .If { a ∈ L | α ( a ) = 0 } 6 = { } , we choose by Zorn’s lemma a maximalorthogonal family in this set. By the separability of L this familyis at most countably infinite. Hence there is a maximal orthogonalsequence ( a i ) i ≥ with α ( a i ) = 0 for all i . Let a = ∨ i a i and observethat α ( a ) = 0 . To establish that a ⊥ = s ( α ) , we show that for any x ∈ L , α ( x ) = 0 if an only if x ⊥ a ⊥ , that is, x ≤ a . If x ≤ a , then This property has independently been introduced in [66] and [34] and it isknown to be equivalent to the fact that each α ∈ S has a (unique) support in L [3,Theorem 11.4.3]. Our proof is an adaption of the corresponding results in [3]. Another sourceleading to this conclusion is given by the results of Section 2.5.2 of [56]. ( x ) ≤ α ( a ) = 0 . To show the converse, assume that α ( x ) = 0 . By the(dual) Jauch-Piron property there is a c ∈ L such that x ≤ c, a ≤ c and α ( c ) = 0 . If x a then c = a (since otherwise a = c ≥ x ) and thus, byorthomodularity c = a ∨ ( c ∧ a ⊥ ) . Therefore, α ( c ) = α ( a ) + α ( c ∧ a ⊥ ) and thus α ( c ∧ a ⊥ ) = 0 . Since c ∧ a ⊥ is orthogonal to each a i we mayexpand the maximal orthogonal sequence ( a i ) , which is a contradiction.Hence, x ≤ a , showing that a ⊥ = s ( α ) .We show next that each a ∈ L , a = 0 , is the support of some α ∈ S .Let W ( a ) = { x ∈ L | x = s ( α ) for some α such that α ( a ) = 1 } . By thesufficiency of P this is a nonempty set. Moreover, if x ∈ W ( a ) , x = s ( α ) and α ( a ) = 1 then x ≤ a . Let ( x i ) be a maximal (countable) orthogonalsequence in W ( a ) and define b = ∨ i x i (so that b ≤ a ). As above, if b = a , then a ∧ b ⊥ would be an element in W ( a ) pairwise orthogonalwith each x i , which is not possible. Thus ∨ i x i = a . Since any x i is thesupport of some α i (for which α i ( a ) = 1 ), then a is the support of allthe convex combinations P w i x i (with all w i > ).It remains to be shown that L is a complete lattice. Let a, b ∈ L , a = 0 = b (if a or b is 0 the supremum and infimum exist trivially).Let α, β ∈ S be such that s ( α ) = a, s ( β ) = b , and consider the state γ = λα + (1 − λ ) β for some = λ = 1 . Clearly s ( γ ) = a ∨ b . ByDe Morgan laws one gets the dual result. It is well-known that everyseparable orthomodular σ -orthocomplete lattice is complete, see e.g.[56, Lemma 2.5.2f]. (cid:3) There are three further important properties the logic ( S , L ) mustpossess in order to provide a geometric representation of the elementsof L as subspaces of a vector space. The first property is the atomicity: L is atomic if every a ∈ L , a = 0 , contains an atom. We recall that anelement p ∈ L is an atom if for any a ∈ L , a = 0 , the condition a ≤ p implies a = p . We let At( L ) denote the set of atoms in L .To get the atomicity of L we pose the following assumption concern-ing the identification of pure states. In Section 4 this assumption isformulated in terms of operations and it forms a part of the projectionpostulate. Identification of pure states : Let α ∈ P . For any β ∈ S , if β ( s ( α )) = 1 then β = α . Proposition 1.
With the assumptions of Theorem 3, the identificationof pure states implies that the support of any pure state is an atom.Moreover, L is atomic and the map P ∋ α s ( α ) ∈ At( L ) is abijection. roof. Let p = s ( α ) be the support of α ∈ P and let a ∈ L , a = 0 , be such that a ≤ p . Since a = 0 there is a β ∈ P such that β ( a ) = 1 .From a ≤ p one then gets β ( p ) = 1 , which means that β = α . Since a ≥ s ( β ) and s ( β ) = s ( α ) = p , one has a ≥ p and thus a = p , that is, p is an atom.For a ∈ L , a = 0 , there is a α ∈ P such that α ( a ) = 1 . Therefore s ( α ) ≤ a , showing that L is atomic.We leave it as an exercise to show that the mapping P ∋ α → s ( α ) ∈ At( L ) is injective and surjective. (cid:3) The second ingredient required to establish the vector space realiza-tion is the covering property : for any a ∈ L and p ∈ At( L ) , if a ∧ p = 0 ,then a ∨ p covers a , that is, for any b ∈ L , if a ≤ b ≤ a ∨ p , then b = a or b = a ∨ p . Since L is an atomic lattice the covering property canequivalently be formalized as follows: for any a ∈ L , p ∈ At( L ) , theelement ( a ∨ p ) ∧ a ⊥ is either an atom or [56, Prop. 3.2.17]. To ob-tain the covering property for ( S , L ) we adapt a part of the projectionpostulate reflecting the possibility of actualizing potential propertieswith minimal disturbance . In Section 4 we present a full formulation ofthis postulate together with an elucidation of its physical motivation.The ideality assumption (I1) of a filter given there corresponds to thefollowing minimal disturbance requirement. Minimal disturbance : If α ∈ P , a ∈ L , and α ( a ) = 0 , then thereexists a pure state β ∈ P such that β ( a ) = 1 , that is, s ( β ) ≤ a ,and α ( s ( β )) = α ( a ) . Proposition 2. (Bugajska, Bugajski, [6] ) With the assumptions ofTheorem 3 and the identification of the pure states, the minimal dis-turbance implies the covering property.
Proof.
Let p ∈ At( L ) , a ∈ L and p = s ( α ) . Let α and α be thepure states such that α ( a ) = α ( s ( α )) and α ( a ⊥ ) = α ( s ( α )) as givenby the minimal disturbance. Clearly, α ( s ( α ) ∨ s ( α )) = 1 , so that p ≤ s ( α ) ∨ s ( α ) and p ∨ a ⊥ ≤ s ( α ) ∨ a ⊥ . Hence ( p ∨ a ⊥ ) ∧ a ≤ ( s ( α ) ∨ a ⊥ ) ∧ a = s ( α ) , which means that ( p ∨ a ⊥ ) ∧ a is either theatom s ( α ) or , that is, the covering property holds in L . (cid:3) We collect the above assumptions concerning the set of states of thelogic in the form of an axiom.
Axiom 6.
The set S of states of the logic ( S , L ) , with a separable L ,has a sufficient set of pure states, the Jauch-Piron property, and itallows the identification of pure states and the minimal disturbance. n atomic lattice with covering property is often referred to as anAC lattice. We may thus conclude that Axioms 5 and 6 imply that L is a complete orthomodular AC lattice and that the support functiongives a bijective correspondence between the sets P and At( L ) .The final ingredient required to establish a geometric representationof ( S , L ) is the irreducibility of L .3.3. The classical case excluded.
There are various features of quan-tum mechanicsthat have been elevated to the status of fundamentalprinciples of the theory. These include the notions of superposition,complementarity, uncertainty, entanglement, nonunique decomposabil-ity and purification of mixed states, and irreducibility of probabilities,which are unquestionably among the most widely discussed character-istic traits of quantum mechanics. In Remark 1 the unique decompos-ability of mixed states into its pure components is seen to be closelyrelated to a classical description and it goes hand in hand with theBoolean structure of the set of decision effects. Here we discuss brieflythe notions of superposition and complementarity to show that for aproper quantum system the logic L is far from being Boolean. We candraw on the full structure of the pair ( S , L ) introduced in the preced-ing subsections 3.1 and 3.2 even though not all of it is actually neededhere. Superpositions.
There are several formulations of the notion of super-position in quantum logic. We adopt the following definition takenfrom [64, p. 53] as a formalization of the intuitive ideas of Dirac [17]:
Definition 3.
A pure state α ∈ P is a superposition of pure states α , α ∈ P if and only if α ( a ) = α ( a ) = 1 implies α ( a ) = 1 for every a ∈ L . Equivalently, a pure state α is a superposition of pure states α and α if and only if s ( α ) ≤ s ( α ) ∨ s ( α ) . Instead of stating directly a superposition principle we give the fol-lowing definition:A physical system with the structure ( S , L ) is a proper quantumsystem if for every two pure states α, β ∈ P , α = β , there existsa third one γ ∈ P , α = γ = β , which is their superposition.It is then a simple, but important consequence that the logic of a properquantum system is irreducible , that is, the centre of L Cent( L ) = { c ∈ L | a = ( a ∧ c ) ∨ ( a ∧ c ⊥ ) for any a ∈ L } contains only the trivial elements and e . (For a proof, see e.g. [56,Cor. 3.2.4].) emark 3. Note that if c ∈ Cent( L ) , then for any a ∈ L , there is anobservable E ∈ e O such that c = E ( X ) and a = E ( Y ) for some value sets X and Y . Thus for a proper quantum system there is no (nontrivial)proposition (or property) that could be measured together with evryother proposition (or property). By contrast, if Cent( L ) = L , the set L forms a Boolean σ -algebra for which the theorems of Stone [62] andLoomis [40] and Sikorski [61] give a representation as a σ -algebra A ofsubsets of a set Ω . Remark 4.
If a pure state α is a superposition of pure states β and γ ,then also β is a superposition of α and γ , and likewise γ is a superposi-tion of α and β . This is the exchange property and it is often includedin the notion of superposition of states. In the present context thisproperty is equivalent to the covering property, obtained above fromthe projection postulate. For a proof, see e.g. [56, Prop. 3.2.17]. Complementarity.
The existence of pairs of complementary observablesis another fundamental feature of quantum mechanics. Following theideas of Bohr [5] we say that two observables are complementary ifall the experimental arrangements which unambiguously define theseobservables are mutually exclusive. Again, there are various ways offormalizing this intuitive idea. We adopt the following definition ap-propriate to the logic ( S , L ) . Definition 4.
Properties a, b ∈ L are complementary if they are dis-joint, that is, a ∧ b = 0 , but not orthogonal, that is, a b ⊥ . Equiv-alently, a, b ∈ L are complementary , if for any α ∈ S , the condition α ( a ) = 1 implies = α ( b ) = 1 , and the condition α ( b ) = 1 implies = α ( a ) = 1 . In a Boolean logic, the conditions a ∧ b = 0 and a ≤ b ⊥ are equivalent.This means that if there are complementary properties in L then L cannot be Boolean.It is a simple exercise to show that L is irreducible if for any a ∈ L , a = 0 , e , there is a b ∈ L such that a and b are complementary. As analternative to the previous definition, we could call a physical systemwith the structure ( S , L ) a proper quantum system if for any a ∈ L , a = 0 , e , there is a b ∈ L such that a and b are complementary. It thenfollows that for a proper quantum system the logic L is irreducible.We summarize the main result of this section. Corollary 1.
Let ( S , O , p ) be the statistical duality satisfying Axioms 1and 3. If the logic ( S , L ) defined by a (nonempty) subset e O of observ-ables satisfies Axioms 5 and 6 and the physical system in question is a roper quantum system, then L is an irreducible complete orthomodularAC lattice and there is a bijective correspondence (given by the supportfunction) between the set P of pure states in S and the set At( L ) ofatoms in L . Filters and the projection postulate
Assume now that the statistical duality satifies the stronger Axioms 2and 4. The existence of a sufficiently large subset of observables e O ⊂ O leading to the fundamental result of Corollary 1 goes together with theexistence of a subset e O of operations such that e ◦ e O = L . This suggeststhat the same conclusion could be reached by singling out a sufficientlylarge and regular set of operations. This is what we consider next. Remark 5.
With the structure specified by Theorem 3 and Proposi-tions 1 and 2 one may construct for each a ∈ L a map φ a : S → S whose restriction on pure states is uniquely defined by s ( φ a ( α )) =( s ( α ) ∨ a ⊥ ) ∧ a , α ∈ P , and which has the typical properties of a statetransformation caused by an ideal first kind measurement [54, 55, 8].Due to the properties of the support function s : S → L the map φ a fails to be linear, that is, it is not an operation in the sense of Sec-tion 2.4. Apart from this the characteristic properties of such a φ a serve below as the defining properties of filters.4.1. Filters.
Filters are a special kind of operations reflecting certainideality properties that the so-called yes-no (or simple ) measurementsmay or may not possess. The properties of filters have been discussedextensively in the literature, see, e.g. [54, 55, 15, 16, 19, 20, 47, 8, 3, 37].This allows us to be brief in their introduction. The properties of filtersare defined through their action on pure states. The definition thuspresumes that the set P = ex( S ) of pure states is not empty.An operation φ ∈ O is pure if(P1) φ ( α ) ∈ [0 , · P for any pure state α ∈ P ,and an operation φ is of the first kind if(F1) e ( φ ( α )) = 1 implies φ ( α ) = α for any α ∈ P ,(F2) e ( φ ( α )) = e ( φ ( α )) for any α ∈ P .To define the ideality of an operation we first assume that any purestate can be identified by an operation:(S1) for any pure state α ∈ P there is a unique φ α ∈ O such that e ( φ α ( β )) = 1 implies β = α for any β ∈ P .We then say that a pure operation φ is ideal if I1) e ( φ ( α )) = e ( φ α ′ ( α )) for any α ∈ P , with φ ( α ) = 0 , where α ′ = e ( φ ( α )) − φ ( α ) and φ α ′ as in (S1).A pure, ideal, first kind operation is a filter and we let O f denote theset of filters. We comment briefly on the defining properties of filters.The purity (P1) of an operation means that it takes a pure state toa pure state with a possible loss in the intensity. As pure states maybe interpreted as maximal information states, a pure operation leavesthe system in a maximal information state whenever it was in such astate.With the so-called ideality assumptions one usually aims at mini-mizing the influence on the state caused by an operation performed onthe system. In addition to the purity condition (P1) and the first kindconditions (F1) and (F2), the condition (I1) aims at that. It claimsthat an ideal φ maps any pure state α onto an eigenstate of φ closest to α , thus disturbing the system to a minimal extent. This is the minimaldisturbance assumption of Section 3.2.Of the two first kind conditions (F1) and (F2), (F1) claims that if φ does not lead to a detectable effect when performed on the system in apure state α then, provided that the operation is good enough, it doesnot alter the state of the system, either. According to (F2), a repeatedapplication of a good operation does not lead to a new effect.As an immediate consequence of the defining properties of filters,we note that they are not only weakly repeatable ( e ( φ ( α )) = e ( φ ( α )) for any α ∈ P ) but also repeatable ( φ ( α ) = φ ( α ) for any α ∈ P ).Moreover, filters satisfy the most common ideality requirement: if agood operation φ is performed on the system in a pure state α whichis an eigenstate of a good operation φ (i.e. e ( φ ( α )) = e ( α ) ) whichcommutes weakly with φ (i.e. φ ◦ φ and φ ◦ φ are isotonic), then φ leaves the system in a state which is still an eigenstate of φ .We say that the set O f of filters is sufficiently rich if the operationsof (S1) are filters and(S2) for each filter φ ∈ O f there is another filter φ ′ ∈ O f suchthat e ◦ φ ′ = ( e ◦ φ ) ⊥ .Condition (S1), the identification of pure states, expresses the commonbelief that any pure state α can be produced by a particular selection orfiltering process φ α , which under the conditions (F1) and (F2) receivesthe form φ α ( β ) = e ( φ α ( β )) α for any β ∈ P . The second sufficiencycondition (S2) stipulates that if an effect a can be obtained from afilter, that is a = e ◦ φ for some φ ∈ O f , then also its ‘negation’ a ⊥ = e − a can be produced by a pure ideal first kind operation. .2. Projection postulate.
The set L of propositions of the convexscheme ( S , O , E ) is now defined as the set of all decision effects a ∈ ex( E ) with nonempty certainly-yes-domain a = { α ∈ P | a ( α ) = 1 } together with the null effect , L = { a ∈ ex( E ) | a = 0 or a = ∅} . For a given system ( S , O , E ) the set O f of filters may be emptyand the set L of propositions may be trivial { , e } . However, for any φ ∈ O f , φ = 0 , the resulting effect e ◦ φ has a nonempty certainly-yes-domain ( e ◦ φ ) . By Remark 5 it is also natural to expect thatfor any a ∈ L , a = 0 , there is a filter φ a such that e ◦ φ a = a . Withthe projection postulate we confirm this expectation together with aunicity assumption. In that we also assume that the set of pure statesis not only nonempty but is also strongly ordering on L , that is, it isordering and for any f, g ∈ L , if f = ∅ , and f ⊆ g , then f ≤ g . Axiom 7.
The statistical duality ( S , O , p ) of Axioms 2 and 4 satisfiesthe projection postulate if the set P of pure states is strongly orderingon L , the subset of filters O f ⊂ O is sufficiently rich and there is abijective mapping Φ : L → O f with the property: a ( α ) = e (Φ( a )( α )) for every a ∈ L and α ∈ P . The projection postulate guarantees the existence of a sufficientlyrich collection of the important class of operations associated with thepure, ideal, first-kind measurements, but it does not restrict the the-ory to such measurements only. Neither does it distinguish betweenclassical and quantum descriptions. In any case, this postulate hasstrong structural implications on the order structure of the set L ofpropositions. They will be studied next. Lemma 1.
For a statistical duality ( S , O , p ) satisfying the projectionpostulate, the set L of propositions is a nonempty partially ordered setwith a a ⊥ as orthocomplementation. Proof.
Since P = ∅ the set O f of filters is nonempty and thus also L = ∅ . The set ex( E ) of decision effects is closed under the map a a ⊥ = e − a . If a ∈ L , with a = e ◦ Φ( a ) , Φ( a ) ∈ O f , then by(S2) a ⊥ = e ◦ Φ( a ) ′ for some Φ( a ) ′ ∈ O f , so that a ⊥ ∈ L . Clearly, Φ( a ) ′ = Φ( a ⊥ ) . Let b ∈ L be such that b ≤ a and b ≤ a ⊥ , and assumethat b = 0 . Then for any α ∈ b , a ( α ) = 1 and a ⊥ ( α ) = 1 which isimpossible. Thus b = 0 , that is, a ∧ L a ⊥ = 0 . The remaining claimsare obvious. (cid:3) emma 2. For any a, b ∈ L , if a ≤ b ⊥ , then a + b ∈ L . Similarly, forany triple ( a, b, c ) of mutually orthogonal elements of L , a + b, a + c, b + c, a + b + c ∈ L . Proof.
For a ⊥ b , a + b ≤ e . Now ( a + b ) ⊇ a ∪ b = ∅ . Assume that a + b = ( f + g ) for some f, g ∈ E . Then ( a + b ) = f ∩ g so that a + b ≤ f and a + b ≤ g . Thus a + b = a + b + ( f − ( a + b ) + g − ( a + b )) , which implies that f = g = a + b , that is, a + b ∈ ex( E ) . Hence a + b ∈ L .Copying the argument for a triple of mutually orthogonal elements a, b, c ∈ L one immediately concludes also that a + b + c ∈ L . (cid:3) Corollary 2.
For a statistical duality ( S , O , p ) satisfying the projectionpostulate, the set L of propositions is orthomodular. Proof.
Let a, b, c ∈ L be a triple of mutually orthogonal elements.Then not only a + b, a + c and b + c but also a + b + c ∈ L . Thismeans that L is triangle-closed in the sense of [39]. By [39, Theorem3.2] this is equivalent to L ⊆ E being orthomodular, and, in particular, a + b = a ∨ L b for a, b ∈ L , a ≤ b ⊥ . (cid:3) Lemma 3.
For any α ∈ P , e ◦ φ α ∈ At( L ) . Moreover, L is atomic,that is, any a ∈ L , a = 0 , contains an atom. The map P ∋ α e ◦ φ α ∈ At( L ) is a bijection, with e ◦ φ α being the support of α . Proof.
Let a ∈ L , α ∈ P , and assume that a ≤ Φ − ( φ α ) . If a = 0 thenfor any β ∈ a , Φ − ( φ α )( β ) = e ( φ α ( β )) = 1 , so that by (S1) β = α ,that is a = { α } . Therefore Φ( a ) = φ α , or equivalently, a = Φ − ( φ α ) ,which entails that for any α ∈ P , Φ − ( φ α ) is an atom. Clearly, for any a ∈ L , one has Φ − ( φ α ) = e ◦ φ α ≤ a for all α ∈ a . (cid:3) Lemma 4.
The set P of pure states is sufficient for L , and L has theJauch-Piron property. Proof.
The sufficiency of P for L is obvious. Let a, b ∈ L be such that a ∩ b = ∅ . For any α ∈ a ∩ b , e ◦ φ α is contained both in a and in b and e ( φ α α ) = 1 . (cid:3) We observe that the range E ( A ) of an observable E ∈ O is Booleanif it is contained in L . Therefore we may again justify the separabilityassumption of L with the requirement that any Boolean subsystem of L could be realized as the range of an observable. With the separabilityassumption, L thus acquires the structure specified in Theorem 3. By irtue of Proposition 2, the ideality property (I1) of filters then givesthe covering property. Hence we have the following. Theorem 4.
If the operational description ( S , O , E ) defined by Axioms2 and 4 satifies the projection postulate and L = { a ∈ ex( E ) | a = ∅ or a = 0 } is separable, then L forms a complete atomic orthomodularorthocomplemented lattice with the covering property. Moreover, thesupport function gives a bijective correspondence between the pure statesin S and the atoms of L . To get the irreducibility of L it is most straightforward to requirethat any two pure states can be superposed into a new pure state. Withthe structures given by the projection postulate we may immediatelyadopt Definition 3 to conclude that for a proper quantum system thestructures of Corollary 1 are again available.It is to be emphasized, however, that even though the two sets ofaxioms [1,3,5,6] and [2,4,7] lead to the common structure of Corollary 1,the first approach starts with the weaker assumptions concerning thepair ( S , E ) . Therefore, it is concievable that there are pairs ( S , L ) with the structure of this corollary appearing as models for the firstapproach but not for the second approach.5. Hilbert space coordinatization
The basic problem.
In the Hilbert space formulation of quan-tum mechanics the pair ( S , E ) is given as the sets of density operatorsand effect operators on a complex separable Hilbert space, whereas L isidentified as the set of (orthogonal) projections on it. It is a deep theo-rem of Gleason [27] which assures that all the probability measures on L arise from the density operators through the familiar trace formula. Untill now we have presented two sets of axioms for the structures ( S , E ) and ( S , L ) associated with a proper quantum system. The re-maining problem of this axiomatic approach is to show that the onlyrealization of this abstract structure is the one given by the Hilbertspace quantum mechanics. In the following we present an outline ofthe solution of this problem, including some still open, critical points.The traditional way of approaching the problem has been to isolatefirst the structure of L and to look for the models of this structurealone. Then, only after having obtained the models of L , the structureof S is added and E is determined. One might expect that this way ofvoluntarily neglecting a good part of the basic structures of the pairs For a detailed discussion of this theorem, see, e.g. [64] S , E ) and ( S , L ) cannot be the most optimal approach. We return tothis question later.5.2. The fundamental representation theorem.
Let K be a di-vision ring with an involutive antiautomorphism λ λ ∗ (such that ( λ + µ ) ∗ = λ ∗ + µ ∗ , ( λµ ) ∗ = µ ∗ λ ∗ , λ ∗∗ = λ ) and let V be a (left) vectorspace over K . A Hermitian form on V is a mapping f : V × V → K with the following properties: for any u, v, w ∈ V, λ, µ ∈ K , f ( λu + µv, w ) = λf ( u, w ) + µf ( v, w ) f ( u, v ) ∗ = f ( v, u ) f ( v, v ) = 0 implies v = 0 . If V admits a Hermitian form f we say that V , or rather ( V, K, ∗ , f ) , isa Hermitian space . A subspace M ⊂ V of a Hermitian space is f - closed if M = M ⊥⊥ , where M ⊥ = { v ∈ V | f ( v, x ) = 0 for all x ∈ M } . Let L f ( V ) denote the set of all f -closed subspaces of V . In additionto the trivial subspaces { } and V any finite dimensional subspace is f -closed. Clearly, if V is infinite dimensional they do not exhaust the set L f ( V ) . The subset inclusion ⊆ together with the map M M ⊥ give L f ( V ) the structure of an irreducible complete orthocomplemented AClattice. The converse result is a fundamental representation theoremof projective geometry, proved in detail, for instance, in [45, Theorem34.5]: Theorem 5. If L is an irreducible complete orthocomplemented AClattice of lenght at least 4, that is, the lenght of a maximal chain is ≥ , then there is a Hermitian space ( V, K, ∗ , f ) such that L is ortho-isomorphic to the lattice L f ( V ) . A Hermitian space ( V, K, ∗ , f ) is orthomodular if for any M ∈ L f ( V ) , M + M ⊥ = V. A Hermitian space ( V, K, ∗ , f ) is known to be orthomodular if and onlyif the lattice L f ( V ) is orthomodular, see e.g. [52, Theorem 2.8]. Thuswe have the following corollary: Corollary 3.
Assume that L is an irreducible complete orthocomple-mented orthomodular AC lattice of length at least 4. Then there isan orthomodular space ( V, K, ∗ , f ) such that L is ortho-isomorphic to L f ( V ) , in short, L ≃ L f ( V ) . In particular, all the finite dimensional ubspaces of V are in L f ( V ) and the atoms of L f ( V ) are the one-dimensional subspaces of V . The pure states α ∈ P are in one-to-one onto correspondence withthe atoms [ v ] = { λv | λ ∈ K } ∈ L f ( V ) and they are uniquely deter-mined by their values on the atoms, that is, by the numbers α [ v ] ([ u ]) ∈ [0 , , [ u ] ∈ At( L f ( V )) . It is to be stressed that this corollary does notyet give any information on the structure of the real numbers α [ v ] ([ u ]) ;in particular, it is not known if α [ v ] ([ u ]) could be related to the K -number f ( u ′ , v ′ ) for some v ′ ∈ [ v ] , u ′ ∈ [ u ] . If such a conclusion couldbe reached then K should be an extension of R .The well-known models for an orthomodular space ( V, K, ∗ , f ) are theclassical Hilbert spaces H over R , C , or H , the quaternions. In thesemodels, the form f is the scalar product on V and by Gleason’s theoremthe probabilities α [ v ] ([ u ]) are of the form α [ v ] ([ u ]) = | f ( v ′ , u ′ ) | for any v ′ ∈ [ v ] , u ′ ∈ [ u ] with f ( v ′ , v ′ ) = f ( u ′ , u ′ ) = 1 , provided that dim( H ) ≥ which is the case in Corollary 3. However, the Hilbert spaces do notexhaust the orthomodular spaces. In the finite dimensional case this isevident, as shown by a simple example. Example 1.
Any finite dimensional Hermitian space ( V, K, ∗ , f ) is or-thomodular and each subspace M of V is f -closed, see, e.g. [32]. Hencethe lattice L f ( V ) of f -closed subspaces coincides with the lattice L ( V ) of all subspaces of V , which is modular (and thus also orthomodular).It is obvious that the space ( V, K, ∗ , f ) need not be a Hilbert space. Towitness, consider the finite-dimensional rational vector space Q n withthe natural form f ( q , p ) = P ni =1 q i p i . The form f is Hermitian so that ( Q n , Q , id , f ) is an orthomodular space. Clearly Q n is not completewith respect to the distance defined by f . We return to this examplein Sect. 5.4 where we study probability measures on L f ( Q n ) .This example leaves open the infinite dimensional case. In his sem-inal paper [35] Keller was able to construct an explicit example of aninfinite dimensional orthomodular space that is very far from being aHilbert space. Further examples emerged later [28], and we now knowthat there are plenty of orthomodular spaces other than the classicalHilbert spaces. The problem then arises to characterize the Hilbertspaces among the orthomodular spaces. This is solved in the nextsubsection.5.3. A theorem of Solér. heorem 6. [Solér, [60] ] Let ( V, K, ∗ , f ) be an orthomodular space.The division ring K is either R , C or H and ( V, K, ∗ , f ) is the corre-sponding Hilbert space if and only if there is an infinite sequence ofnonzero vectors e i , i = 1 , , · · · such that f ( e i , e j ) = 0 for all i = j ,with the property f ( e i , e i ) = f ( e j , e j ) for all i, j . This remarkable result characterizes the Hilbert space models of theorthomodular spaces in a, perhaps, unexpected way. We emphasizethat in this theorem V is required to be infinite dimensional and or-thomodular . The next two examples demonstrate that neither of theseassumptions can be relaxed. Example 2.
The vectors (1 , , . . . , , . . . (0 , . . . , , form an orthonor-mal basis in ( Q n , Q , id , f ) but the space, though orthomodular, is nota Hilbert space. Example 3.
Consider the infinite dimensional vector space V = ℓ ( Q ) of the square summable sequencies of rational numbers q = ( q , q , q , . . . ) with the Hermitian form f ( q , p ) = P ∞ i =1 q i p i . The lattice L f ( V ) of f -closed subspaces is a complete, irreducible AC lattice of infi-nite lenght but it is not orthomodular. The vectors (1 , , . . . , , . . . ) . . . (0 , . . . , , , , . . . ) . . . form an orthonormal basis in V which is not aHilbert space.For L ≃ L f ( V ) the existence of a sequence of mutually orthogo-nal vectors ( e i ) in V follows from the assumption that L containsan infinite sequence of pairwise orthogonal atoms. Such an assump-tion is physically well motivated e.g. by the spectroscopic data or bythe assumption that the quantum system can be localized in an Eu-clidean space. It is then worth stressing that, contrary to our intuitionthat comes from using complex numbers, it is the ‘norm’ requirement f ( e i , e i ) = f ( e j , e j ) that is here highly non-trivial. Indeed, suppose that f ( e i , e i ) = λ and f ( e i , e j ) = 0 . We have to find an element µ ∈ K suchthat µf ( e j , e j ) µ ∗ = λ ; in this way f ( µe j , µe j ) = f ( e i , e i ) , see Sect. 5.4.This is a quadratic equation in K that cannot be solved in general. In R or C one would simply take the square root of the positive number λλ ∗ whereas e.g. in Q this would not work. For instance, a one-dimensionalsubspace [ q ] = { λ q | λ ∈ Q } of ( Q n , Q , id , f ) contains a unit vector onlyif pP q i is rational.Combining Corollary 3 with the theorem of Solér we get the follow-ing: If K = R then ∗ is the identity. For K = C the map ∗ cannot be the identityand if it is continuous then it is the complex conjugation. For K = H the map isthe quaternionic conjugation. heorem 7. Assume that L is an irreducible complete orthocomple-mented orthomodular AC lattice that contains an infinite sequence oforthogonal atoms. Then there is an orthomodular space ( V, K, ∗ , f ) suchthat L is orthoisomorphic to L f ( V ) . K is R , C , or H and ( V, K, ∗ , f ) is the corresponding Hilbert space if and only if V contains an infinitesequence of mutually orthogonal vectors ( v i ) with the property (7) f ( v i , v i ) = f ( v j , v j ) for all i, j. By assumption, there is an infinite sequence of orthogonal vectors.The essential question is which properties of ( S , L ) would imply thatsuch a sequence could be chosen to have the ‘norm’ property (7).Purely lattice theoretical conditions on L are known that are suffi-cient to ensure that L ≃ L ( H ) for a Hilbert space H . We can referto the so-called ’angle bisection property’ [50] or the existence of ’har-monic conjugate’ pairs of atoms [65, 32]. They are of geometric natureand, in the light of the present understanding, they seem to lack anyphysical interpretation. Therefore, they are not useful for the axiomaticscheme followed here.The necessary and sufficient conditions for the conclusion L ≃ L ( H ) of Theorem 7 are expressed in terms of ( V, f ) . One might expect thatthe assumptions of this theorem together with the full structure of thepair ( S , L ) , in particular, the bijection between P and At( L ) , couldalready force L to be a Hilbertian lattice. We investigate some aspectsof this question in Sect. 6, although to the best of our knowledge, thisproblem remains still largely open.The other two remaining questions are: what can be said if L hasonly a finite length, and how can the states be represented, once wehave represented L .As concerns the latter question we recall that if ( V, K, ∗ , f ) is a clas-sical Hilbert space of dimension at least 3, then all the probabilitymeasures on L = L f ( V ) are described by Gleason’s theorem. Accord-ing to it, for any probability measure α on L f ( V ) , there is a uniquepositive trace one operator ρ : V → V such that, for any M ∈ L f ( V ) ,we have α ( M ) = tr (cid:2) ρP M (cid:3) , where P M is the projection onto M . For dim( V ) = 2 , the set of all probability measures on L f ( V ) is, however,much bigger than those defined by the density operators. But these ad-ditional probability measures are not supported by L f ( V ) ; for details,see [3, Sect. 25.2]. To the best of our knowledge, there is no empiricalevidence which would require the use of such probability measures asstates of a two-level quantum system.The situation is very different when ( V, K, ∗ , f ) is not a classicalHilbert space. Very little is known of the probability measures on he lattices L f ( V ) . Keller [36] gives examples of nonclassical L f ( V ) for which one may construct a rich supply of probability measures m : L f ( V ) → [0 , , see also [28, Problem 7]. No classification the-orem of the Gleason type is available for these examples, and it alsoseems that for them there is no one-to-one correspondence betweenpure probability measures and atoms of L f ( V ) .5.4. Finite dimensional case: an example.
As already noted above,the structure of a quantum logic ( S , L ) satisfying L ≃ L f ( V ) with dim( V ) < ∞ may be substantially different from the infinite dimen-sional case. To emphasize this further we continue Example 1 withdetermining the set of states S for the logic L f ( Q n ) .Consider the rational orthomodular space Q n with the lattice L ( Q n ) = L f ( Q n ) . For any M ∈ L ( Q n ) one has Q n = M + M ⊥ . Hence foreach q ∈ Q n there is a unique decomposition q = q + q , with q ∈ M, q ∈ M ⊥ . This entails that the map P M : Q n → Q n defined by P M q = q is linear, idempotent and Hermitian, that is, f ( q , P M p ) = f ( P M q , p ) for all q , p ∈ Q n . For any atom [ v ] ∈ L ( Q n ) ,one may thus define the map α [ v ] , with(8) α [ v ] ( M ) = f ( v , P M v ) f ( v , v ) , which is a probability measure on L ( Q n ) and its support is the definingatom, that is, s ( α [ v ] ) = [ v ] . Clearly, the mapping α [ v ] s ( α [ v ] ) gives aone-to-one correspondence between the set of probability measures on L ( Q n ) of the form α [ v ] and the set of atoms of L ( Q n ) .Let P at be the set of states defined by the atoms of L ( Q n ) , that is, α ∈ P at if α = α [ v ] for some v ∈ Q n , v = 0 . Any σ -convex combinationof states ( α [ v i ] ) i ≥ with weights ( λ i ) i ≥ is again a state (probabilitymeasure) on L ( Q n ) . We let S at denote the set of all such states. Ithas all the regularity properties of Section 3.2, including the strongordering on L ( Q n ) . First of all, each α ∈ S at has a support in L ( Q n ) ;if α = P i λ i α [ v i ] , then s ( α ) = ∨{ [ v i ] | λ i = 0 } . Moreover, if s ( α ) = [ v ] for some atom [ v ] , then α = α [ v ] . Secondly, ex( S at ) = P at , which alsoconfirms that the restriction of the support projection to P at defines abijection between the sets ex( S at ) and At( L ( Q n )) .Let S denote the set of all probability measures on L ( Q n ) . Wedemonstrate next that S at is a proper subset of S .To begin with, we note first that L ( Q n ) can be naturally embedded in L ( R n ) . Indeed, for M ∈ L ( Q n ) , choose an orthogonal basis e , . . . , e k , k ≤ n , with M = span Q { e , . . . , e k } , and define f M = span R { e , . . . , e k } .Then L ( Q n ) ∋ M f M ∈ L ( R n ) is an injective mapping. ix a nonzero vector v ∈ R n such that at least one of its componentsis irrational. For M ∈ L ( Q n ) define(9) α [ v ] ( M ) = (cid:10) v | P f M v (cid:11) h v | v i , where h · | · i denotes the natural inner product in R n . Clearly, α [ v ] isa probability measure on L ( Q n ) . However, for any q ∈ Q n , q = 0 ,α [ v ] ([ q ]) = 0 , which shows that α [ v ] has not support in L ( Q n ) . Hence α [ v ] is not in S at .This example shows that the subspace lattice L ( Q n ) of the non-Hilbertian orthomodular space ( Q n , Q , id , f ) admits a rich subset ofstates S at that has all the listed regularity properties. Therefore, onecould consider ( S at , L ( Q n )) as a logic of a proper quantum system. Inthis case the logic admits also additional probability measures whichcannot be considered as states of the quantum system since they arenot supported in L ( Q n ) .One may speculate whether ( V, K, ∗ , f ) can be forced to be a classicalHilbert space by requiring that the set of all pure probability measuresis defined on L f ( V ) so as to be in one-to-one correspondence with theatoms of L f ( V ) . Although this seems to be an appealing property, itremains a conjecture for now, or rather, a hope for the future.6. The role of symmetries in the representation theorem
In his authoritative review [32] Holland formulated the axiom of am-ple unitary group according to which for each pair of mutually orthog-onal vectors u, v ∈ V there is a bijective linear map U : V → V suchthat U ( v ) = u and f ( U x, U y ) = f ( x, y ) for each x, y ∈ V . Clearly, thisassumption does the job. However, this is a very strong assumption,and, in any case, it is not a property given by the pair ( S , L ) . Ratherthan accepting this postulate we follow [10] to elucidate the physicalcontent hidden in such an axiom.6.1. Implementing symmetries as operators on V . The idea ofsymmetry receives its natural mathematical representation as a trans-formation on the set of entities the symmetry refers to. The basicstructures are now coded in the sets L and S and in the duality be-tween them. These sets possess various physically relevant structureswhich define the corresponding automorphism groups. From the out-set any of them could be used to formulate the notion of symmetryin quantum logic. In view of the theorem of Solér we shall consider nly two of them: symmetries of the set At( L ) of atoms of L , and thesymmetries of the logic L . Definition 5. a) A mapping ℓ o : At( L ) → At( L ) is an orthosymmetry if it is bijective and for any p, q ∈ At( L ) , p ⊥ q ⇐⇒ ℓ o ( p ) ⊥ ℓ o ( q ) . b) A mapping ℓ : L → L is a symmetry if it is bijective and it preservesthe order and the orthocomplementation, that is, for any a, b ∈ L , a ≤ b ⇐⇒ ℓ ( a ) ≤ ℓ ( b ) ℓ ( a ⊥ ) = ℓ ( a ) ⊥ . Let
Aut o (At( L )) and Aut( L ) denote the sets of orthosymmetriesand symmetries on L , respectively. Both of these sets are groups withrespect to the composition of mappings. Moreover, any symmetry ℓ ,when restricted to At( L ) , defines an orthosymmetry.Assume now that the logic L allows a vector space coordinatizationof the form L ≃ L f ( V ) for an orthomodular space ( V, K, ∗ , f ) . Any ℓ o ∈ Aut o (At( L )) as well as ℓ ∈ Aut( L ) defines the correspondingautomorphism on the set P ( V ) of atoms of L f ( V ) and on the whole L f ( V ) , respectively. We continue to denote them as ℓ o and ℓ and callthem orthosymmetry and symmetry, respectively. Lemma 5.
Let ℓ o ∈ Aut o ( P ( V )) . There is a unique symmetry ˆ ℓ o ∈ Aut( L f ( V )) such that ˆ ℓ o ([ v ]) = ℓ o ([ v ]) for all [ v ] ∈ P ( V ) . Moreover,the map Aut o ( P ( V )) ∋ ℓ o ˆ ℓ o ∈ Aut( L f ( V )) is a group isomorphism. Proof.
Let ℓ o ∈ Aut o (At( L f ( V ))) and define for any (nonempty)subset M ⊆ V , M = { } , ˆ ℓ o ( M ) = { x ∈ ℓ o ([ v ]) | v ∈ M, v = 0 } and put ˆ ℓ o ( { } ) = { } . Since Aut o (At( L f ( V ))) is a group we alsohave c ℓ − o defined in the same way. A direct computation shows that ˆ ℓ o ( c ℓ − o ( M )) = KM and c ℓ − o (ˆ ℓ o ( M )) = KM . Indeed, for any (nonempty) M , M = { } , we have c ℓ − o (ˆ ℓ o ( M )) = { w ∈ ℓ − ([ v ]) | v ∈ ˆ ℓ o ( M ) } = { w ∈ ℓ − ([ v ]) | v ∈ ℓ o ([ x ]) , x ∈ M, x = 0 } = { w ∈ ℓ − ( ℓ o ([ x ])) | x ∈ M, x = 0 } = { w ∈ [ x ] | x ∈ M, x = 0 } = KM, Various definitions of the notion of symmetry in quantum mechanics are studiede.g. in [9, 49]. nd if M = { } , then c ℓ − o ( ˆ ℓ o ( { } ) = { } . Similarly, one gets the otherset equality. If M is a subspace, then ˆ ℓ o ( c ℓ − o ( M )) = M , c ℓ − o (ˆ ℓ o ( M )) = M. Using the fact that for any two (nonzero) vectors u, v ∈ V and forany ℓ o ∈ Aut o (At( L f ( V ))) , f ( u, v ) = 0 is equivalent to [ u ] ⊥ [ v ] andto ℓ o ([ u ]) ⊥ ℓ o ([ v ]) (meaning that f ( x, y ) = 0 for any x ∈ ℓ o ([ u ]) , y ∈ ℓ o ([ v ]) ) one easily verifies that ˆ ℓ o ( M ) ⊥ = ˆ ℓ o ( M ⊥ ) for any (nonempty) set M ⊂ V .Let now M ∈ L f ( V ) . Since M = ( M ⊥ ) ⊥ , we have ˆ ℓ o ( M ) = ˆ ℓ o ( M ⊥ ) ⊥ so that by [64, Lemma 4.35] ˆ ℓ o ( M ) is an f -closed subspace of V , thatis, ˆ ℓ o ( M ) ∈ L f ( V ) . Hence, the map L f ( V ) ∋ M ˆ ℓ o ( M ) ∈ L f ( V ) iswell-defined. Clearly, it is a bijection, with the inverse (ˆ ℓ o ) − = c ℓ − o ,it preserves the orthocomplementation and, by construction, also theorder. Therefore, for any ℓ o ∈ Aut o (At( L f ( V ))) , ˆ ℓ o ∈ Aut( L f ( V )) .For any v ∈ V, v = 0 , ˆ ℓ o ([ v ]) = ℓ o ([ v ]) , which shows that ˆ ℓ o ex-tends the map ℓ o . Let ℓ ∈ Aut( L f ( V )) and assume that it is anotherextension of ℓ o . Since the lattice L f ( V ) is atomistic we now have ℓ ( M ) = ℓ ( ∨{ [ v ] | [ v ] ⊆ M } ) = ∨{ ℓ ([ v ]) | [ v ] ⊆ M } = ∨{ ℓ o ([ v ]) | [ v ] ⊆ M } = ∨{ ˆ ℓ o ([ v ]) | [ v ] ⊆ M } = ˆ ℓ o ( ∨{ [ v ] | [ v ] ⊆ M } ) = ˆ ℓ o ( M ) for any M ∈ L f ( V ) , showing that ℓ = ˆ ℓ o , that is, the extension isunique. The map Aut o (At( L f ( V ))) ∋ ℓ o ˆ ℓ o ∈ Aut( L f ( V )) is thuswell-defined. Its injectivity and surjectivity are obvious and it alsopreserves the group structure: c ℓ − o = ( ˆ ℓ o ) − and \ ℓ o ◦ ℓ ′ o = ˆ ℓ o ◦ ˆ ℓ ′ o for all ℓ o , ℓ ′ o ∈ Aut o (At( L f ( V ))) . (cid:3) Let L ( V ) be the (complete, irreducible, modular, AC) lattice of allsubspaces of V and let Aut( L ( V )) be the group of order isomorphismson L ( V ) . Let F ( V ) = { L ∈ L ( V ) | dim( L ) < ∞} and recall that P ( V ) ⊆ F ( V ) ⊆ L f ( V ) . Note also that any M ∈ L ( V ) can be ex-pressed as M = ∪{ L ∈ F ( V ) | L ⊆ M } = ∨{ L ∈ F ( V ) | L ⊆ M } , and ℓ ( L ) ∈ F ( V ) for any L ∈ F ( V ) , ℓ ∈ Aut( L f ( V )) . Lemma 6.
For any ℓ ∈ Aut( L f ( V ))Φ ℓ ( M ) = ∪{ ℓ ( L ) | L ⊆ M, L ∈ F ( V ) } efines an order-preserving bijection Φ ℓ : L ( V ) → L ( V ) which extendsthe map ℓ . Proof.
This is an adoptation of the proof of [24, Lemma 1]. We showfirst that Φ ℓ ( M ) ∈ L ( V ) for any M ∈ L ( V ) . Indeed, if x ∈ Φ ℓ ( M ) , then x ∈ ℓ ( L ) for some L ∈ F ( V ) , L ⊆ M , and thus λx ∈ ℓ ( L ) ⊆ Φ ℓ ( M ) for any λ ∈ K . Moreover, if y ∈ Φ ℓ ( M ) , then y ∈ ℓ ( H ) for some H ∈ F ( V ) , H ⊆ M , and thus x + y ∈ ℓ ( L ) + ℓ ( H ) = ℓ ( L ) ∨ ℓ ( H )= ℓ ( L ∨ H ) = ℓ ( L + H ) ⊆ Φ ℓ ( M ) , since the subspaces involved are all finite dimensional. Hence Φ ℓ ( M ) ∈ L ( V ) for any M ∈ L ( V ) .To prove that the map Φ ℓ has an inverse, we need the followingobservation: { H = ℓ ( L ) | L ∈ F ( V ) , L ⊆ M } = { H ∈ F ( V ) | H ⊆ Φ ℓ ( M ) } . Since ℓ ∈ Aut( L f ( V )) is a group, we also have Φ ℓ − ( M ) = ∪{ ℓ − ( L ) | L ⊆ M, L ∈ F ( V ) } , and using the above observation one quickly confirms that Φ ℓ − (Φ ℓ ( M )) = Φ ℓ (Φ ℓ − ( M )) for any M ∈ L ( V ) . Hence, for any ℓ ∈ Aut( L f ( V )) the map Φ ℓ : L ( V ) → L ( V ) is a bijection, with the inverse Φ ℓ − . By definitionthe map Φ ℓ preserves the order, that is, Φ ℓ ( M ) ⊆ Φ ℓ ( N ) , if and onlyif M ⊆ N for any M, N ∈ L ( V ) . Hence, Φ ℓ ∈ Aut( L ( V )) for any ℓ ∈ Aut( L f ( V )) We show next that Φ ℓ extends ℓ . Let M ∈ L f ( V ) ⊆ L ( V ) . Since Φ ℓ ( M ) = ∪{ ℓ ( L ) | L ∈ F ( V ) , L ⊆ M } , ℓ ( M ) = ∪{ H | H ∈ F ( V ) , H ⊆ ℓ ( M ) } , and H ⊆ ℓ ( M ) if and only if H = ℓ ( F ) , F ∈ F ( V ) , F ⊆ M , weobserve that Φ ℓ ( M ) = ℓ ( M ) . Since any M ∈ L ( V ) can be expressedas M = ∨{ L | L ∈ F ( V ) , L ⊂ M } one easily verifies that Φ ℓ is the onlyorder isomorphism of L ( V ) which extends ℓ ∈ Aut( L f ( V )) . (cid:3) Let S : V → V be a bijective map which is g -linear, that is, S is bi-jective and additive on V , g an isomorphism of K , and S ( λv ) = g ( λ ) Sv for all v ∈ V and λ ∈ K . Such an S induces an order isomorphismon L ( V ) by Φ S ( M ) = { Sv | v ∈ M } , and if S ′ is another bijective h -linear map V → V inducing the same order isomorphism, that is, Φ S = Φ S ′ , then there is a λ ∈ K such that S v = λS ′ v for any v ∈ V [2, Proposition III.1.2, Corollary III.1.2]. The first fundamental the-orem of projective geometry [2, p. 44] gives the converse result: if im( V ) ≥ , then for any Φ ∈ Aut( L ( V )) there is an isomorphism g : K → K and a bijective g -linear map S : V → V inducing Φ .Let ℓ o ∈ Aut o ( P ( V )) and let S ℓ : V → V thus be a bijective g ℓ -linearmap which induces the extension Φ ℓ ∈ Aut( L ( V )) of the extension ℓ ∈ Aut( L f ( V )) of ℓ o . Thus, for any v ∈ V, v = 0 , S ℓ ( Kv ) = Φ ℓ ( Kv ) = ℓ ( Kv ) = ℓ o ( Kv ) . Since ℓ o preserves the orthogonality of atoms, the one-dimensional sub-spaces S ℓ ( Ku ) and S ℓ ( Kv ) are orthogonal if and only if Ku and Kv areorthogonal, that is, f ( Ku, Kv ) = 0 if and only if f ( S ℓ ( Ku ) , S ℓ ( Kv )) =0 . A direct computation shows that the map ( u, v ) g − ℓ ( f ( S ℓ u, S ℓ v )) =: ˜ f ( u, v ) is a Hermitian form on ( V, K, ∗ ) such that ˜ f ( u, v ) = 0 if and only if f ( u, v ) = 0 for all u, v ∈ V . By virtue of the infinite dimensionalversion of the Birkhoff - von Neumann theorem [45] there is a nonzero ρ ℓ ∈ K such that ˜ f ( u, v ) = ρ ℓ f ( u, v ) for all u, v ∈ V . Moreover, ρ ℓ isa symmetric element of K , and since λ λ ∗ is an antiautomorphismof K one also has λρ ℓ = ρ ℓ λ for all λ ∈ K , that is, ρ ℓ ∈ Cent( K ) , thecentre of K . Corollary 4.
For any ℓ o ∈ Aut o ( P ( V )) there is an isomorphism g ℓ : K → K and a bijective orthogonality preserving g ℓ -linear map S ℓ : V → V such that (10) ℓ o ( Kv ) = S ℓ ( Kv ) for any v ∈ V , v = 0 . Moreover, there is a ρ ℓ ∈ Cent( K ) , ρ ℓ = 0 , ρ ℓ = ρ ∗ ℓ , such that (11) f ( S ℓ u, S ℓ v ) = g ℓ ( ρ ℓ ) g ℓ ( f ( u, v )) for all v, u ∈ V . This corollary is a precursor of the theorem of Wigner according towhich the ‘transition probability preserving bijections on the set of purestates’ are implemented by unitary or antiunitary operators acting onthe underlying Hilbert space H of the standard logic L = L ( H ) [64]. Inthat frame, the orthosymmetries are exactly the transition probabilityzero preserving bijections on the pure states. If dim( H ) ≥ thenthis group coincides with the group of transition probability preservingbijections on the set of pure states [9, Corollary 4]. Now the length of L is at least 4 so that dim( V ) ≥ . .2. Symmetries and the Solér conditions.
We now study the roleof symmetry in providing a partial justification of the assumptions ofSolér’s theorem. Clearly, the result is obtained if L ≃ L f ( V ) has thefollowing property: Given any two mutually orthogonal atoms [ x ] , [ y ] ∈ L f ( V ) , there are nonzero vectors x ′ ∈ [ x ] and y ′ ∈ [ y ] such that(12) f ( x ′ , x ′ ) = f ( y ′ , y ′ ) . Before investigating the conditions the theorem of Solér imposeson the set of symmetries, we recall that a proper quantum object isan elementary quantum object with respect to a group G of (for in-stance, space-time) transformations if there is a group homomorphism σ : G → Aut o ( P ( V )) and if for any pure state (atom) [ v ] ∈ P ( V ) , theset { σ g ([ v ]) | g ∈ G } of pure states (atoms) is complete in the sense ofsuperpositions, that is, any other pure state (atom) [ u ] ∈ P ( V ) can beexpressed as a superposition of some of the pure states (atoms) σ g ([ v ]) , g ∈ G . Even though this does not solve our problem, it shows thatfor an elementary quantum object the set of symmetries Aut o ( P ( V )) is rather large and the notion of superposition has a role in it. Thenext lemma binds the above condition (12) more tightly to the issue athand. Lemma 7.
Let [ x ] , [ y ] be any two mutually orthogonal atoms in L f ( V ) .If there are nozero vectors x ′ ∈ [ x ] and y ′ ∈ [ y ] such that f ( x ′ , x ′ ) = f ( y ′ , y ′ ) then there is an ℓ o ∈ Aut o ( P ( V )) which swaps [ x ] and [ y ] , thatis, ℓ o ([ x ]) = [ y ] and ℓ o ([ y ]) = [ x ] . Moreover, there is a [ v ] ≤ [ x ] ∨ [ y ] such that ℓ o ([ v ]) = [ v ] . Proof.
Let M = [ x ] ∨ [ y ] = [ x ] ⊕ [ y ] . Clearly, [ x ] = [ x ′ ] , [ y ] = [ y ′ ] . Any u ∈ M can be written uniquely as u = αx ′ + βy ′ , α, β ∈ K . Fix λ ∈ Cent( K ) , λ = 0 , and define U M ( u ) = U M ( αx ′ + βy ′ ) = λ ( αy ′ + βx ′ ) . The map U M is a linear bijection on M , and for any u, v ∈ M , λf ( u, v ) λ ∗ = f ( U M u, U M v ) . Let v = x ′ + y ′ and observe that [ v ] is fixed by U M . Since M is f -closed, V = M + M ⊥ , so that any w ∈ V can uniquely be decomposed as w = w + w , with w ∈ M, w ∈ M ⊥ . We define a canonical extension of U M to the whole V by U w = U ( w + w ) = U M w + λw . Then U is a bijective linear mapon V . Moreover, f ( U w, U v ) = λf ( w, v ) λ ∗ for all w, v ∈ V , and foreach u ∈ M , U u = U M u . Hence, in particular, Φ U ([ x ]) = [ y ] , Φ U ([ y ]) = [ x ] , Φ U ([ v ]) = [ v ] . (cid:3) his lemma shows that condition (12) implies the existence of aspecial symmetry of L f ( V ) that interchanges the two orthogonal atoms [ x ] and [ y ] and has a superposition of them as a fixed point.To get the opposite implication, and thus come to the final conclu-sion, we add the following two assumptions, the first concerning thegroup Aut o ( P ( V )) , the second the form f :( A ) The symmetry group is abundant in the following sense:for any pair of mutually orthogonal atoms [ x ] , [ y ] ∈ P ( V ) thereis a symmetry ℓ o ∈ Aut o ( P ( V )) that swaps [ x ] and [ y ] , thatis, ℓ o ([ x ]) = [ y ] and ℓ o ([ y ]) = [ x ] , and has some of their su-perpositions as a fixed point, that is, ℓ o ([ v ]) = [ v ] for some [ v ] ≤ [ x ] ∨ [ y ] ;( R ) The form f is regular in the following sense: for each v ∈ V , f ( v, v ) ∈ Cent( K ) , and g ( f ( v, v )) = f ( v, v ) for anyautomorphism g of K . Lemma 8.
Let [ x ] , [ y ] be any two mutually orthogonal atoms in L f ( V ) .If the group Aut o ( P ( V )) is abundant and the form f is regular thenthere are nonzero vectors x ′ ∈ [ x ] and y ′ ∈ [ y ] such that f ( x ′ , x ′ ) = f ( y ′ , y ′ ) . Proof.
Let ℓ o ∈ Aut o ( P ( V )) be an orthosymmetry swapping the atoms [ x ] and [ y ] and having a [ v ] ≤ [ x ] ∨ [ y ] as a fixed point. Let S ℓ , g ℓ , ρ ℓ constitute a realization of ℓ o as given in Corollary 4. Applying Eq.(11) first to the vector v and its transform S ℓ v = λv , λ ∈ K , one gets g ℓ ( ρ ℓ ) = λλ ∗ . Applying then the same equation to x and S ℓ x = αy, α ∈ K , one gets f ( αy, αy ) = g ℓ ( ρ ℓ ) g ℓ ( f ( x, x )) = λf ( x, x ) λ ∗ = f ( λx, λx ) which completes the proof. (cid:3) We summarize the results of this section in the form of a theorem.
Theorem 8.
Assume that the logic ( S , L ) of the statistical duality ( S , O , p ) has the structure of Corollary 1. Assume that the system hasan abundant set of orthosymmetries. If there is an infinite sequence ofmutually orthogonal atoms in L , and if the form f of the coordinatiza-tion ( V, K, ∗ , f ) of the logic is regular, then V is a Hilbert space over R , C , or H , and L is (ortho-order) isomorphic with the lattice of closedsubspaces of the Hilbert space V . ith this theorem the statistical duality ( S , O , p ) of a proper quan-tum system is completely resolved: the states α ∈ S of the systemare identified with positive trace one operators ρ of an infinite dimen-sional classical Hilbert space H , the observables ( E , Ω , A ) ∈ O areexpressed as semispectral measures, also called normalized positive op-erator measures, taking values in the set of bounded operators on H ,and the numbers p ( α, E , X ) are determined to be given by the ‘Bornrule’ p ( α, E , X ) = tr (cid:2) ρ E ( X ) (cid:3) . The pure states are the one-dimensionalprojections and the L -valued observables are the spectral measures.We are left with the question whether the regularity of the form f ,the requirement ( R ), can be stated as a property of the logic ( S , L ) of the duality ( S , O , p ) . Another open question is the choice of thenumber field left open by Theorem 6. We close our paper with a shortcomment on this.It is well known that the complex Hilbert space H is in many respectssimpler than the real or quaternionic Hilbert spaces. We recall only thepowerfull polarization identity (valid in the complex case) and the for-mulation of the Stone theorem which is of fundamental importance.But is the choice C only a mathematical convenience? Some of thedifferences between the three cases have been discussed already in [3,Chapter 22]. In addition to that we mention here the work of Pulman-nová [53] where a symmetry argument is given to rule out the real andquaternionic choices for K . Finally we note that the cases of Hilbertspaces over C and R can be distinguished in terms of the different lowerbounds obtained in the respective derivations of Heisenberg-Kennard-Robertson -type preparation uncertainty relations [38]. Acknowledgement.
We are grateful to Drs Paul Busch and MaciejM¸aczynski for their valuable comments in earlier versions of this man-uscript.
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Pacific J. Math. (19961) 1151-1169. Department of Physics, University of Genova, and INFN Sezione diGenova, Genoa, Italy
E-mail address : [email protected] Turku Centre for Quantum Physics, Department of Physics and As-tronomy, University of Turku, Turku, Finland
E-mail address : [email protected]@utu.fi