Mathematically, quantum mechanics can be regarded as a non-classicalprobability calculus resting upon a non-classical propositional logic.More specifically, in quantum mechanics each probability-bearingproposition of the form “the value of physical quantity \(A\)lies in the range \(B\)” is represented by a projection operatoron a Hilbert space \(\mathbf{H}\). These form a non-Boolean—inparticular, non-distributive—orthocomplemented lattice.Quantum-mechanical states correspond exactly to probability measures(suitably defined) on this lattice.

What are we to make of this? Some have argued that the empiricalsuccess of quantum mechanics calls for a revolution in logic itself.This view is associated with the demand for a realistic interpretationof quantum mechanics, i.e., one not grounded in any primitive notionof measurement. Against this, there is a long tradition ofinterpreting quantum mechanics operationally, that is, as beingprecisely a theory of measurement. On this latter view, it is notsurprising that a “logic” of measurement-outcomes, in asetting where not all measurements are compatible, should prove not tobe Boolean. Rather, the mystery is why it should have theparticular non-Boolean structure that it does in quantummechanics. A substantial literature has grown up around the programmeof giving some independent motivation for thisstructure—ideally, by deriving it from more primitive andplausible axioms governing a generalized probability theory.

• 1. Quantum Mechanics as a Probability Calculus
  • 1.1 Quantum Probability in a Nutshell
  • 1.2 The “Logic” of Projections
  • 1.3 Probability Measures and Gleason’s Theorem
  • 1.4 The Reconstruction of QM
• 2. Interpretations of Quantum Logic
  • 2.1 Realist Quantum Logic
  • 2.2 Operational Quantum Logic
• 3. Generalized Probability Theory
  • 3.1 Discrete Classical Probability Theory
  • 3.2 Test Spaces
  • 3.3 Kolmogorovian Probability Theory
  • 3.4 Quantum Probability Theory
• 4. Logics associated with probabilistic models
  • 4.1 Operational Logics
  • 4.2 Orthocoherence
  • 4.3 Lattices of Properties
• 5. Piron’s Theorem
  • 5.1 Conditioning and the Covering Law
• 6. Classical Representations
  • 6.1 Classical Embeddings
  • 6.2 Contextual Hidden Variables
• 7. Composite Systems
  • 7.1 The Foulis-Randall Example
  • 7.2 Aerts’ Theorem
  • 7.3 Ramifications
• 8. Effect Algebras
  • 8.1 Quantum effects and Naimark’s Theorem
  • 8.2 Sequential Effect Algebras
• Bibliography
• Academic Tools
• Other Internet Resources
• Related Entries

1. Quantum Mechanics as a Probability Calculus

It is uncontroversial (though remarkable) that the formal apparatus ofquantum mechanics reduces neatly to a generalization of classicalprobability in which the role played by a Boolean algebra of events inthe latter is taken over by the “quantum logic” ofprojection operators on a Hilbert space.^[1] Moreover, the usual statistical interpretation of quantum mechanicsasks us to take this generalized quantum probability theory quiteliterally—that is, not as merely a formal analogue of itsclassical counterpart, but as a genuine doctrine of chances. In thissection, I survey this quantum probability theory and its supportingquantum logic.^[2]

[For further background on Hilbert spaces, see the entry onquantum mechanics. For further background on ordered sets and lattices, see thesupplementary document:The Basic Theory of Ordering Relations. Concepts and results explained these supplements will be used freelyin what follows.]

1.1 Quantum Probability in a Nutshell

The quantum-probabilistic formalism, as developed by von Neumann[1932], assumes that each physical system is associated with a(separable) Hilbert space \(\mathbf{H}\), the unit vectors of whichcorrespond to possible physicalstates of the system. Each“observable” real-valued random quantity is represented bya self-adjoint operator \(A\) on \(\mathbf{H}\), the spectrum of whichis the set of possible values of \(A\). If \(u\) is a unit vector inthe domain of \(A\), representing a state, then the expected value ofthe observable represented by \(A\) in this state is given by theinner product \(\langle Au,u\rangle\). The observables represented bytwo operators \(A\) and \(B\) are commensurable iff \(A\) and \(B\)commute, i.e.,AB = BA. (For further discussion, see theentry on quantum mechanics.)

1.2 The “Logic” of Projections

As stressed by von Neumann, the \(\{0,1\}\)-valued observables may beregarded as encoding propositions about—or, to use his phrasing,properties of—the state of the system. It is not difficult toshow that a self-adjoint operator \(P\) with spectrum contained in thetwo-point set \(\{0,1\}\) must be a projection; i.e., \(P^2 = P\).Such operators are in one-to-one correspondence with the closedsubspaces of \(\mathbf{H}\). Indeed, if \(P\) is a projection, itsrange is closed, and any closed subspace is the range of a uniqueprojection. If \(u\) is any unit vector, then \(\langle Pu,u\rangle =\llvert Pu\rrvert ^2\) is the expected value of the correspondingobservable in the state represented by \(u\). Since this observable is\(\{0,1\}\)-valued, we can interpret this expected value as theprobability that a measurement of the observable will producethe “affirmative” answer 1. In particular, the affirmativeanswer will have probability 1 if and only ifPu = u; thatis, \(u\) lies in the range of \(P\). Von Neumann concludes that

… the relation between the properties of a physical system onthe one hand, and the projections on the other, makes possible a sortof logical calculus with these. However, in contrast to the conceptsof ordinary logic, this system is extended by the concept of“simultaneous decidability” which is characteristic forquantum mechanics. (1932: 253)

Let’s examine this “logical calculus” ofprojections. Ordered by set-inclusion, the closed subspaces of\(\mathbf{H}\) form a complete lattice, in which the meet (greatestlower bound) of a set of subspaces is their intersection, while theirjoin (least upper bound) is the closed span of their union. Since atypical closed subspace has infinitely many complementary closedsubspaces, this lattice is not distributive; however, it isorthocomplemented by the mapping

In view of theabove-mentioned one-one correspondence between closed subspaces andprojections, we may impose upon the set \(L(\mathbf{H})\) thestructure of a complete orthocomplemented lattice, defining \(P\leQ\), where \(\rran (P) \subseteq \rran (Q)\) and \(P' = 1 - P\) (sothat \(\rran (P') = \rran (P)^{\bot})\). It is straightforward that\(P\le Q\) just in case \(PQ = QP = P\). More generally, ifPQ =QP, then \(PQ = P\wedge Q\), the meet of \(P\) and \(Q\) in\(L(\mathbf{H})\); also in this case their join is given by \(P\vee Q= P+Q - PQ\).

Adhering to the idea that commuting observables—in particular,projections—are simultaneously measurable, we conclude that themembers of a Boolean sub-ortholattice of \(L(\mathbf{H})\) aresimultaneously testable. This suggests that we can maintain aclassical logical interpretation of the meet, join and orthocomplementas applied to commuting projections.

1.3 Probability Measures and Gleason’s Theorem

The foregoing discussion motivates the following. Call projections\(P\) and \(Q\)orthogonal, and write \(P\binbot Q\) iff \(P\le Q'\). Note that \(P\binbot Q\) iff \(PQ = QP = 0\). If \(P\) and\(Q\) are orthogonal projections, then their join is simply their sum;traditionally, this is denoted \(P\oplus Q\). We denote the identitymapping on \(\mathbf{H}\) by \(\mathbf{1}\).

Here is one way in which we can manufacture a probability measure on\(L(\mathbf{H})\). Let \(u\) be a unit vector of \(\mathbf{H}\), andset \(\mu_u (P) = \langle Pu,u\rangle\). This gives the usualquantum-mechanical recipe for the probability that \(P\) will havevalue 1 in the state \(u\). Note that we can also express \(\mu_u\) as\(\mu_u(P) = Tr(P P_u)\), where \(P_u\) is the one-dimensionalprojection associated with the unit vector \(u\), i.e., \(P_u(x) =\langle x, u \rangle u\) for all \(x \in \mathbf{H}\).

More generally, if \(\mu_i, i=1,2,\ldots\), are probability measureson \(L(\mathbf{H})\), then so is any “mixture”, or convexcombination \(\mu = \sum_i t_i\mu_i\) where \(0\le t_i\le 1\) and\(\sum_i t_i = 1\). Given any sequence \(u_1, u_2,\ldots\), of unitvectors, let \(\mu_i = \mu_{u_{ i}}\) and let \(P_i = P_{u_{ i}}\).Forming the operator

one sees that

An operator expressible in this way as a convex combination ofone-dimensional projections in is called adensity operator.Density operators are the standard mathematical representation forgeneral (pure or “mixed”) quantum-mechanical states. Wehave just seen that every density operator \(W\) gives rise to acountably additive probability measure on \(L(\mathbf{H})\). Thefollowing striking converse, due to A. Gleason [1957], shows that thetheory of probability measures on \(L(\mathbf{H})\) is co-extensivewith the theory of (mixed) quantum mechanical states on\(\mathbf{H}\):

An important consequence of Gleason’s Theorem is that\(L(\mathbf{H})\) does not admit any probability measures having onlythe values 0 and 1. To see this, note that for any density operator\(W\), the mapping \(u \rightarrow \langle Wu,u\rangle\) is continuouson the unit sphere of \(\mathbf{H}\). But since the latter isconnected, no continuous function on it can take only the two values 0and 1. This result is often taken to rule out the possibility of“hidden variables”—an issue taken up in more detailin section 6.

1.4 The Reconstruction of QM

From the single premise that the “experimentalpropositions” associated with a physical system are encoded byprojections in the way indicated above, one can reconstruct the restof the formal apparatus of quantum mechanics. The first step, ofcourse, is Gleason’s theorem, which tells us that probabilitymeasures on \(L(\mathbf{H})\) correspond to density operators. Thereremains to recover, e.g., the representation of“observables” by self-adjoint operators, and the dynamics(unitary evolution). The former can be recovered with the help of theSpectral theorem and the latter with the aid of a deep theorem of E.Wigner on the projective representation of groups. See also R. Wright[1980]. A detailed outline of this reconstruction (which involves somedistinctly non-trivial mathematics) can be found in the book ofVaradarajan [1985]. The point to bear in mind is that, once thequantum-logical skeleton \(L(\mathbf{H})\) is in place, the remainingstatistical and dynamical apparatus of quantum mechanics isessentially fixed. In this sense, then, quantum mechanics—or, atany rate, its mathematical framework—reduces to quantumlogic and its attendant probability theory.

2. Interpretations of Quantum Logic

The reduction of QM to probability theory based on \(L(\mathbf{H})\)is mathematically compelling, but what does it tell us aboutQM—or, assuming QM to be a correct and complete physical theory,about the world? How, in other words, are we to interpret the quantumlogic \(L(\mathbf{H})\)? The answer will turn on how we unpack thephrase, freely used above,

• (*) The value of theobservable \(A\) lies in the range \(B\).

One possible reading of (*) isoperational:“measurement of the observable \(A\) would yield (or will yield,or has yielded) a value in the set \(B\)”. On this view,projections represent statements about the possible results ofmeasurements. This sits badly with realists of a certain stripe, who,shunning reference to “measurement”, prefer to understand(*) as aproperty ascription:

the system has a certain categorical property, which corresponds tothe observable \(A\) having, independently of any measurement, a valuein the set \(B\).

(One must be careful in how one understands this last phrase, however:construed incautiously, it seems to posit a hidden-variablesinterpretation of quantum mechanics of just the sort ruled out byGleason’s Theorem. I will have more to say about thisbelow.)

2.1 Realist Quantum Logic

The interpretation of projection operators as representing theproperties of a physical system is already explicit in vonNeumann’sGrundlagen.. However, the logical operationsdiscussed there apply only to commuting projections, which areidentified with simultaneously decidable propositions. In 1936Birkhoff and von Neumann took a step further, proposing to interpretthe lattice-theoretic meet and join of projections as theirconjunction and disjunction,whether or not they commute.Immediately this proposal faces the problem that the lattice\(L(\mathbf{H})\) is not distributive, making it impossible to givethese “quantum” connectives a truth-functionalinterpretation. Undaunted, von Neumann and Birkhoff suggested that theempirical success of quantum mechanics as a framework for physicscasts into doubt the universal validity of the distributive laws ofpropositional logic. Their phrasing remains cautious:

Whereas logicians have usually assumed that properties … ofnegation were the ones least able to withstand a critical analysis,the study of mechanics points to the distributive identities …as the weakest link in the algebra of logic. (1936: 837)

In the 1960s and early 1970s, this thesis was advanced rather moreaggressively by a number of authors, including especially DavidFinkelstein and Hilary Putnam, who argued that quantum mechanicsrequires a revolution in our understanding of logicper se.According to Putnam, “Logic is as empirical as geometry.… We live in a world with a non-classical logic” ([1968]1979: 184).

For Putnam, the elements of \(L(\mathbf{H})\) represent categoricalproperties that an object possesses, or does not, independently ofwhether or not we look. Inasmuch as this picture of physicalproperties is confirmed by the empirical success of quantum mechanics,we must, on this view, accept that the way in which physicalproperties actually hang together is not Boolean. Since logic is, forPutnam, very much the study of how physical properties actually hangtogether, he concludes that classical logic is simply mistaken: thedistributive law is not universally valid.

Classically, if \(S\) is the set of states of a physical system, thenevery subset of \(S\) corresponds to a categorical propertyof the system, and vice versa. In quantum mechanics, the state spaceis the (projective) unit sphere \(S = S(\mathbf{H})\) of a Hilbertspace. However, not all subsets of \(S\) correspond toquantum-mechanical properties of the system. The latter correspondonly to subsets of the special form \(S \cap \mathbf{M}\), for\(\mathbf{M}\) a closed linear subspace of \(\mathbf{H}\). Inparticular, only subsets of this form are assigned probabilities. Thisleaves us with two options. One is to take only these specialproperties as “real” (or “physical”, or“meaningful”), regarding more general subsets of \(S\) ascorresponding to no real categorical properties at all. The other isto regard the “quantum” properties as a small subset ofthe set of all physically (or at any rate, metaphysically) reasonable,but not necessarilyobservable, properties of the system. Onthis latter view, the set ofall properties of a physicalsystem is entirely classical in its logical structure, but we declineto assign probabilities to the non-observable properties.^[3]

This second position, while certainly not inconsistent with realismper se, turns upon a distinction involving a notion of“observation”, “measurement”,“test”, or something of this sort—a notion thatrealists are often at pains to avoid in connection with fundamentalphysical theory. Of course, any realist account of a statisticalphysical theory such as quantum mechanics will ultimately have torender up some explanation of how measurements are supposed to takeplace. That is, it will have to give an account of which physicalinteractions between “object” and “probe”systems count as measurements, and of how these interactions cause theprobe system to evolve into final “outcome-states” thatcorrespond to—and have the same probabilities as—theoutcomes predicted by the theory. This is the notoriousmeasurement problem.

In fact, Putnam advanced his version of quantum-logical realism asoffering a (radical) dissolution of the measurement problem: Accordingto Putnam, the measurement problem (and indeed every otherquantum-mechanical “paradox”) arises through an improperapplication of the distributive law, and hencedisappearsonce this is recognized. This proposal, however, is widely regarded as mistaken.^[4]

As mentioned above, realist interpretations of quantum mechanics mustbe careful in how they construe the phrase “the observable \(A\)has a value in the set \(B\)”. The simplest and most traditionalproposal—often dubbed the “eigenstate-eigenvaluelink” (Fine [1973])—is that (*) holds if and only if ameasurement of \(A\) yields a value in the set \(B\) with certainty,i.e., with (quantum-mechanical!) probability 1. While this certainlygives a realist interpretation of (*),^[5] it does not provide a solution to the measurement problem. Indeed, wecan use it to give a sharp formulation of that problem: even though\(A\) is certain to yield a value in \(B\) when measured, unless thequantum state is an eigenstate of the measured observable \(A\), thesystem does not possess any categorical property corresponding to\(A\)’s having a specific value in the set \(B\). Putnam seemsto assume that a realist interpretation of (*) should consist inassigning to \(A\) some unknown value within \(B\), for which quantummechanics yields a non-trivial probability. However, an attempt tomake such assignments simultaneously for all observables runs afoul ofGleason’s Theorem.^[6]

2.2 Operational Quantum Logic

If we put aside scruples about “measurement” as aprimitive term in physical theory, and accept a principled distinctionbetween “testable” and non-testable properties, then thefact that \(L(\mathbf{H})\) is not Boolean is unremarkable, andcarries no implication about logicper se. Quantum mechanicsis, on this view, a theory about the possible statisticaldistributions of outcomes of certain measurements, and itsnon-classical “logic” simply reflects the fact that notall observable phenomena can be observed simultaneously. Because ofthis, the set of probability-bearing events (or propositions) isless rich than it would be in classical probability theory,and the set of possible statistical distributions, accordingly, lesstightly constrained. That some “non-classical” probabilitydistributions allowed by this theory are actually manifested in natureis perhaps surprising, but in no way requires any deep shift in ourunderstanding of logic or, for that matter, of probability.

This is hardly the last word, however. Having accepted all of theabove, there still remains the question ofwhy the logic ofmeasurement outcomes should have the very special form\(L(\mathbf{H})\), and never anything more general.^[7] This question entertains the idea that the formal structure ofquantum mechanics may beuniquely determined by a smallnumber of reasonable assumptions, together perhaps with certainmanifest regularities in the observed phenomena. This possibility isalready contemplated in von Neumann’sGrundlagen (andalso his later work in continuous geometry), but first becomesexplicit—and programmatic—in the work of George Mackey[1957, 1963]. Mackey presents a sequence of six axioms, framing a veryconservative generalized probability theory, that underwrite theconstruction of a “logic” of experimental propositions,or, in his terminology, “questions”, having the structureof a sigma-orthomodular partially-ordered set (see Section 4 and thesupplement documentThe Basic Theory of Ordering Relations for definitions of these terms). The outstanding problem, for Mackey,was to explain why this posetought to be isomorphic to\(L(\mathbf{H})\):

Almost all modern quantum mechanics is based implicitly or explicitlyon the following assumption, which we shall state as an axiom:

Axiom VII: The partially ordered set of all questions in quantummechanics is isomorphic to the partially ordered set of all closedsubspaces of a separable, infinite dimensional Hilbert space.

This axiom has rather a different character from Axioms I through VI.These all had some degree of physical naturalness and plausibility.Axiom VII seems entirely ad hoc. Why do we make it? Can we justifymaking it? … Ideally, one would like to have a list ofphysically plausible assumptions from which one could deduce AxiomVII. Short of this one would like a list from which one could deduce aset of possibilities for the structure … all but one of whichcould be shown to be inconsistent with suitably planned experiments.[Mackey 1963: 71–72]

Since Mackey’s writing there has grown up an extensive technicalliterature exploring variations on his axiomatic framework in aneffort to supply the missing assumptions. The remainder of thisarticle presents a brief survey of the current state of thisproject.

3. Generalized Probability Theory

Rather than restate Mackey’s axioms verbatim, I shall paraphrasethem in the context of an approach to generalized probability theorydue to D. J. Foulis and C. H. Randall having—among the many moreor less analogous approaches available^[8]—certain advantages of simplicity and flexibility. References for this sectionare Foulis, Greechie, and Rüttimann [1992]; Foulis, Piron andRandall [1983]; Randall and Foulis [1983]; see also Gudder [1989];Wilce [2000b] and Wilce [2009] for surveys.

3.1 Discrete Classical Probability Theory

It will be helpful to begin with a review of classical probabilitytheory. In its simplest formulation, classical probability theorydeals with a (discrete) set \(E\) of mutually exclusive outcomes, asof some measurement, experiment, etc., and with the variousprobability weights that can be defined thereon—thatis, with mappings \(\omega : E \rightarrow[0,1]\) summing to 1 over \(E\).^[9]

Notice that the set \(\Delta(E)\) of all probability weights on \(E\)isconvex, in that, given any sequence\(\omega_1,\omega_2,\ldots\) of probability weights and any sequence\(t_1,t_2,\ldots\) of non-negative real numbers summing to one, theconvex sum or “mixture” \(t_1\omega_1 + t_2\omega_2+\ldots\) (taken pointwise on \(E)\) is again a probability weight.The extreme points of this convex set are exactly the“point-masses” \(\delta(x)\) associated with the outcomes\(x \in E\):

Thus, \(\Delta(E)\) is asimplex:each point \(\omega \in \Delta(E)\) is representable in a unique wayas a convex combination of extreme points, namely:

We alsoneed to recall the concept of arandom variable. If \(E\) isan outcome set and \(V\), some set of “values” (realnumbers, pointer-readings, or what not), a \(V\)-valued randomvariable is simply a mapping \(f : E \rightarrow V\). Theheuristic (but it need only be taken as that) is that one“measures” the random variable \(f\) by“performing” the experiment represented by \(E\) and, uponobtaining the outcome \(x \in E\), recording \(f(x)\) as the measuredvalue. Note that if \(V\) is a set of real numbers, or, moregenerally, a subset of a vector space, we may define theexpectedvalue of \(f\) in a state \(\omega \in \Delta(E)\) by:

3.2 Test Spaces

A very natural direction in which to generalize discrete classicalprobability theory is to allow for a multiplicity of outcome-sets,each representing a different “experiment”. To formalizethis, let us agree that atest space is a non-emptycollection A of non-empty sets \(E,F,\ldots\), each construed as adiscrete outcome-set as in classical probability theory. Each set \(E\in \mathcal{A}\) is called atest. The set \(X = \cup\mathcal{A}\) of all outcomes of all tests belonging to\(\mathcal{A}\) is called theoutcome space of\(\mathcal{A}\). Notice that we allow distinct tests to overlap, i.e.,to have outcomes in common.^[10]

If \(\mathcal{A}\) is a test space with outcome-space \(X\), astate on \(\mathcal{A}\) is a mapping \(\omega : X\rightarrow\) [0,1] such that \(\sum_{x\in E} \omega(x) = 1\) forevery test \(E \in \mathcal{A}\). Thus, a state is a consistentassignment of a probability weight to each test—consistent inthat, where two distinct tests share a common outcome, the stateassigns that outcome the same probability whether it is secured as aresult of one test or the other. (This may be regarded as a normativerequirement on the outcome-identifications implicit in the structureof \(\mathcal{A}\): if outcomes of two tests are not equiprobable inall states, they ought not to be identified.) The set of all states on\(\mathcal{A}\) is denoted by \(\Omega(\mathcal{A})\). This is aconvex set, but in contrast to the situation in discrete classicalprobability theory, it is generally not a simplex.

The concept of a random variable admits several generalizations to thesetting of test spaces. Let us agree that asimple (real-valued)random variable on a test space \(\mathcal{A}\) is a mapping \(f: E \rightarrow \mathbf{R}\) where \(E\) is a test in \(\mathcal{A}\).We define theexpected value of \(f\) in a state \(\omega \in\Omega(\mathcal{A})\) in the obvious way, namely, as the expectedvalue of \(f\) with respect to the probability weight obtained byrestricting \(\omega\) to \(E\) (provided, of course, that thisexpected value exists). One can go on to define more general classesof random variables by taking suitable limits (for details, see Younce[1987]).

In classical probability theory (and especially in classicalstatistics) one usually focuses, not on the set of all possibleprobability weights, but on some designated subset of these (e.g.,those belonging to a given family of distributions). Accordingly, by aprobabilistic model, I mean pair \((\mathcal{A},\Delta)\)consisting of a test space \(\mathcal{A}\) and a designated set ofstates \(\Delta \subseteq \Omega(\mathcal{A})\) on \(\mathcal{A}\).I’ll refer to \(\mathcal{A}\) as thetest space and to\(\Delta\) as thestate space of the model.

I’ll now indicate how this framework can accommodate both theusual measure-theoretic formalism of full-blown classical probabilitytheory and the Hilbert-space formalism of quantum probabilitytheory.

3.3 Kolmogorovian Probability Theory

Let \(S\) be a set, understood for the moment as the state-space of aphysical system, and let \(\Sigma\) be a \(\sigma\)-algebra of subsetsof \(S\). We can regard each partition \(E\) of \(S\) into countablymany pair-wise disjoint \(\Sigma\)-measurable subsets as representinga “coarse-grained” approximation to an imagined perfectexperiment that would reveal the state of the system. Let\(\mathcal{A}_{\Sigma}\) be the test space consisting of all suchpartitions. Note that the outcome set for \(\mathcal{A}_{\Sigma}\) isthe set \(X = \Sigma \setminus \{\varnothing \}\) of non-empty\(\Sigma\)-measurable subsets of \(S\). Evidently, the probabilityweights on \(\mathcal{A}_{\Sigma}\) correspond exactly to thecountably additive probability measures on \(\Sigma\).

3.4 Quantum Probability Theory

Let \(\mathbf{H}\) denote a complex Hilbert space and let\(\mathcal{A}_{\mathbf{H}}\) denote the collection of (unordered)orthonormal bases of \(\mathbf{H}\). Thus, the outcome-space \(X\) of\(\mathcal{A}_{\mathbf{H}}\) will be the unit sphere of\(\mathbf{H}\). Note that if \(u\) is any unit vector of\(\mathbf{H}\) and \(E \in \mathcal{A}_{\mathbf{H}}\) is anyorthonormal basis, we have

Thus, each unit vector of \(\mathbf{H}\) determines a probabilityweight on \(\mathcal{A}_{\mathbf{H}}\). Quantum mechanics asks us totake this literally: any “maximal” discretequantum-mechanical observable is modeled by an orthonormal basis, andany pure quantum mechanical state, by a unit vector in exactly thisway. Conversely, every orthonormal basis and every unit vector areunderstood to correspond to such a measurement and such a state.

Gleason’s theorem can now be invoked to identify the states on\(\mathcal{A}_{\mathbf{H}}\) with the density operators on\(\mathbf{H}\): to each state \(\omega\) in\(\Omega(\mathcal{A}_{\mathbf{H}})\) there corresponds a uniquedensity operator \(W\) such that, for every unit vector \(x\) of\(\mathbf{H}, \omega(x) = \langle Wx,x\rangle = Tr(WP_x), P_x\) beingthe one-dimensional projection associated with \(x\). Conversely, ofcourse, every such density operator defines a unique state by theformula above. We can also represent simple real-valued randomvariables operator-theoretically. Each bounded simple random variable\(f\) gives rise to a bounded self-adjoint operator \(A = \sum_{x\inE} f(x)P_x\). The spectral theorem tells us that every self-adjointoperator on \(\mathbf{H}\) can be obtained by taking suitable limitsof operators of this form.

4. Logics associated with probabilistic models

Associated with any probabilistic model \((\mathcal{A},\Delta)\) areseveral partially ordered sets, each of which has some claim to thestatus of an “empirical logic” associated with the model.In this section, I’ll discuss two: the so-calledoperationallogic \(\Pi(\mathcal{A})\) and theproperty lattice\(\mathbf{L}(\mathcal{A},\Delta)\). Under relatively benign conditionson \(\mathcal{A}\), the former is anorthoalgebra. The latteris always a complete lattice, and under plausible further assumptions,atomic. Moreover, there is a natural order preserving mapping from\(\Pi\) to \(\mathbf{L}\). This is not generally an order-isomorphism,but when it is, we obtain a complete orthomodular lattice, and thuscome a step closer to the projection lattice of a Hilbert space.

4.1 Operational Logics

If \(\mathcal{A}\) is a test space, an \(\mathcal{A}\)-eventis a set of \(\mathcal{A}\)-outcomes that is contained in some test.In other words, an \(\mathcal{A}\)-event is simply an event in theclassical sense for any one of the tests belonging to \(\mathcal{A}\).Now, if \(A\) and \(B\) are two \(\mathcal{A}\)-events, we say that\(A\) and \(B\) areorthogonal, and write \(A\binbot B\), ifthey are disjoint and their union is again an event. We say that twoorthogonal events arecomplements of one another if theirunion is a test. We say that events \(A\) and \(B\) areperspective, and write \(A\sim B\), if they share any commoncomplement. (Notice that any two tests \(E\) and \(F\) areperspective, since they are both complementary to the emptyevent.)

While it is possible to construct perfectly plausible examples of testspaces that are not algebraic, many test spaces that one encounters innature do enjoy this property. In particular, the Borel and quantumtest spaces described in the preceding section are algebraic. The moreimportant point is that, as an axiom, algebraicity is relativelybenign, in the sense that many test spaces can be“completed” to become algebraic. In particular, if everyoutcome has probability greater than 1/2 in at least one state, then\(\mathcal{A}\) is contained in an algebraic test space\(\mathcal{B}\) having the same outcomes and the same states as\(\mathcal{A}\) (see Gudder [1989] for details).

It can be shown^[11] that test space \(\mathcal{A}\) is algebraic if and only if itsatisfies the condition

For all events \(A, B\) of \(\mathcal{A}\), if \(A\sim B\), then anycomplement of \(B\) is a complement of \(A\).

From this, it is not hard to see that, for an algebraic test space\(\mathcal{A}\), the relation \(\sim \) of perspectivity is then anequivalence relation on the set of \(\mathcal{A}\)-events. More thanthis, if \(\mathcal{A}\) is algebraic, then \(\sim \) is acongruence for the partial binary operation of forming unionsof orthogonal events: in other words, for all \(\mathcal{A}\)-events\(A, B\), and \(C, A\sim B\) and \(B\binbot C\) imply that \(A\binbotC\) and \(A\cup C \sim B\cup C\).

Let \(\Pi(\mathcal{A})\) be the set of equivalence classes of\(\mathcal{A}\)-events under perspectivity, and denote the equivalenceclass of an event \(A\) by \(p(A)\); we then have a natural partialbinary operation on \(\Pi(\mathcal{A})\) defined by \(p(A)\oplus p(B)= p(A\cup B)\) for orthogonal events \(A\) and \(B\). Setting 0 :\(=p(\varnothing)\) and 1 :\(= p(E), E\) any member of \(\mathcal{A}\),we obtain a partial-algebraic structure \((\Pi(\mathcal{A}),\oplus,0,1)\), called thelogic of \(\mathcal{A}\). This satisfiesthe following conditions:

1. \(\oplus\) is associative and commutative:
   • If \(a\oplus(b\oplus c)\) is defined, so is \((a\oplus b)\oplusc\), and the two are equal
   • If \(a\oplus b\) is defined, so is \(b\oplus a\), and the two areequal.
2. \(0\oplus a = a\), for every \(a \in \mathbf{L}\)
3. For every \(a \in \mathbf{L}\), there exists a unique \(a' \in\mathbf{L}\) with \(a\oplus a' = 1\)
4. \(a\oplus a\) exists only if \(a = 0\)

We may now define:

Thus, the logic of an algebraic test space is an orthoalgebra. One canshow that, conversely, every orthoalgebra arises as the logic\(\Pi(\mathcal{A})\) of an algebraic test space \(\mathcal{A}\)(Golfin [1988]). Note that non-isomorphic test spaces can haveisomorphic logics.

4.2 Orthocoherence

Any orthoalgebra \(\mathbf{L}\) is partially ordered by the relation\(a\le b\) iff \(b = a\oplus c\) for some \(c\binbot a\). Relative tothis ordering, the mapping \(a\rightarrow a'\) is anorthocomplementation and \(a\binbot b\) iff \(a\le b'\). It can beshown that \(a\oplus b\) is always a minimal upper bound for \(a\) and\(b\), but it is generally not theleast upper bound. Indeed,we have the following (Foulis, Greechie and Ruttimann [1992], Theorem 2.12):

An orthoalgebra satisfying condition (b) is said to beorthocoherent. In other words: an orthoalgebra isorthocoherent if and only if finite pairwise summable subsets of\(\mathbf{L}\) are jointly summable. The lemma tells us that everyorthocoherent orthoalgebra is,inter alia, an orthomodularposet. Conversely, an orthocomplemented poset is orthomodular iff\(a\oplus b = a\vee b\) is defined for all pairs with \(a\le b'\) andthe resulting partial binary operation is associative—in whichcase the resulting structure \((\mathbf{L},\oplus ,0,1)\) is anorthocoherent orthoalgebra, the canonical ordering on which agreeswith the given ordering on \(\mathbf{L}\). Thus, orthomodular posets(the framework for Mackey’s version of quantum logic) areequivalent to orthocoherent orthoalgebras.

A condition related to, but stronger than, orthocoherence is that anypairwisecompatible propositions should be jointlycompatible. This is sometimes calledregularity. Mostnaturally occurring orthomodular lattices and posets are regular. Inparticular, Harding (1996, 1998) has shown that the direct-productdecompositions of any algebraic, relational or topological structurecan be organized in a natural way into a regular orthomodular poset.^[12]

Some version of orthocoherence or regularity was taken by Mackey andmany of his successors as an axiom. (Orthocoherence appears, in aninfinitary form, as Mackey’s axiom V; regularity appears in thedefinition of a partial Boolean algebra in the work of Kochen andSpecker (1965).) However, it is quite easy to construct simple modeltest spaces, having perfectly straightforward—evenclassical—interpretations, the logics of which are notorthocoherent. There has never been given any entirely compellingreason for regarding orthocoherence as an essential feature of allreasonable physical models. Moreover, certain apparently quitewell-motivated constructions that one wants to perform with testspaces tend to destroy orthocoherence (seesection 7).

4.3 Lattices of Properties

The decision to accept measurements and their outcomes as primitiveconcepts in our description of physical systems does not mean that wemust forgo talk of the physical properties of such a system. Indeed,such talk is readily accommodated in the present formalism.^[13] In the approach we have been pursuing, a physical system isrepresented by a probabilistic model \((\mathcal{A},\Delta)\), and thesystem’s states are identified with the probability weights in\(\Delta\). Classically,any subset \(\Gamma\) of thestate-space \(\Delta\) corresponds to a categorical property of thesystem. However, in quantum mechanics, and indeed even classically,not every such property will be testable (or “physical”).In quantum mechanics, only subsets of the state-space corresponding toclosed subspaces of the Hilbert space are testable; in classicalmechanics, one usually takes only, e.g., Borel sets to correspond totestable properties: the difference is that the testable properties inthe latter case happen still to form a Boolean algebra of sets, wherein the former case, they do not.

One way to frame this distinction is as follows. Thesupportof a set of states \(\Gamma \subseteq \Delta\) is the set

of outcomes that are possible when the property \(\Gamma\) obtains.There is a sense in which two properties are empiricallyindistinguishable if they have the same support: we cannot distinguishbetween them by means of a single execution of a single test. We mighttherefore wish to identify physical properties with classes ofphysically indistinguishable classical properties, or, equivalently,with their associated supports. However, if we wish to adhere to theprogramme of representing physical properties as subsets (rather thanas equivalence-classes of subsets) of the state-space, we can do so,as follows. Define a mapping \(F : \wp(X) \rightarrow \wp(\Delta)\) by\(F(J) = \{\omega \in \Delta \mid S(\omega) \subseteq J \}\). Themapping \(\Gamma \rightarrow F(S(\Gamma))\) is then aclosureoperator on \(\wp(\Delta)\), and the collection of closed sets(that is, the range of \(F)\) is a complete lattice of sets, closedunder arbitrary intersection.^[14] Evidently, classical properties—subsets of\(\Delta\)—have the same support iff they have the same closure,so we may identify physical properties with closed subsets of thestate-space:

We now have two different “logics” associated with a probabilistic model \((\mathcal{A},\Delta)\) with \(\mathcal{A}\) algebraic: a“logic” \(\Pi(\mathcal{A})\) of experimental propositionsthat is an orthoalgebra, but generally not a lattice, and a“logic” \(\mathbf{L}(\mathcal{A},\Delta)\) of propertiesthat is a complete lattice, but rarely orthocomplemented in anynatural way (Randall and Foulis [1983]). The two are connected by anatural mapping [ ] : \(\Pi \rightarrow \mathbf{L}\), given by\(p \rightarrow[p] = F(J_p)\) where for each \(p\in \Pi\), \(J_p =\{x\in X \mid p(x) \nleq p' \}\). That is, \(J_p\) is the set ofoutcomes that are consistent with \(p\), and [\(p\)] is the largest(i.e., weakest) physical property making \(p\) certain to be confirmedif tested.

The mapping \(p \rightarrow[p\)] is order preserving. For both theclassical and quantum-mechanical models considered above, it is infact an order-isomorphism. Whenever this is the case, \(\Pi\) willinherit from \(\mathbf{L}\) the structure of a complete lattice, whichwill then automatically be orthomodular by Lemma 4.3. In other words,in such cases we have onlyone logic, which is a completeorthomodular lattice. While it is surely too much to expect that [ ]will be an order-isomorphism everyconceivable physicalsystem—indeed, we can easily construct toy examples to thecontrary—the condition is at least reasonably transparent in itsmeaning.

5. Piron’s Theorem

Suppose that the logic and property lattices of a modelareisomorphic, so that the logic of propositions/properties is a completeorthomodular lattice. The question then arises: how close does thisbring us to quantum mechanics—that is, to the projection lattice\(L(\mathbf{H})\) of a Hilbert space?

The answer is: without additional assumptions, not very. The lattice\(L(\mathbf{H})\) has several quite special order-theoretic features.First it isatomic—every element is the join of minimalnon-zero elements (i.e., one-dimensional subspaces). Second, it isirreducible—it can not be expressed as a non-trivialdirect product of simpler OMLs.^[16] Finally, and most significantly, it satisfies the so-calledatomic covering law: if \(p \in L(\mathbf{H})\) is an atomand \(p\nleq q\), then \(p \vee q\)covers \(q\) (no elementof \(L(\mathbf{H})\) lies strictly between \(p \vee q\) and\(q)\).

These properties still do not quite suffice to capture\(L(\mathbf{H})\), but they do get us into the right ballpark. Let\(\mathbf{V}\) be any inner product space over an involutive divisionring \(D\). A subspace \(\mathbf{M}\) of \(\mathbf{V}\) is said to be\(\bot\)-closed iff \(\mathbf{M} = \mathbf{M}^{\bot \bot}\),where \(\mathbf{M}^{\bot} = \{v\in \mathbf{V} \mid \forall m\in\mathbf{M}( \langle v,m\rangle = 0)\}\). Ordered by set-inclusion, thecollection \(L(\mathbf{V})\) of all \(\bot\)-closed subspaces of\(\mathbf{V}\) forms a complete atomic lattice, orthocomplemented bythe mapping \(\mathbf{M} \rightarrow \mathbf{M}^{\bot}\). A theorem ofAmemiya and Araki (1966) shows that a real, complex or quaternionicinner product space \(\mathbf{V}\) with \(L(\mathbf{V})\)orthomodular, is necessarily complete. For this reason, an innerproduct space \(\mathbf{V}\) over an involutive division ring iscalled ageneralized Hilbert space if itslattice\(L(\mathbf{V})\) of \(\bot\)-closed subspaces is orthomodular.The following representation theorem is due to C. Piron [1964]:

It should be noted that generalized Hilbert spaces have beenconstructed over fairly exotic division rings.^[17] Thus, while it brings us tantalizingly close, Piron’s theoremdoes not quite bring us all the way back to orthodox quantummechanics.

5.1 Conditioning and the Covering Law

Let us call a complete orthomodular lattice satisfying the hypothesesof Piron’s theorem aPiron lattice. Can we give anygeneral reason for supposing that the logic/property lattice of aphysical system (one for which these are isomorphic) is a Pironlattice? Or, failing this, can we at least ascribe some clear physicalcontent to these assumptions? The atomicity of \(L\) follows if weassume that every pure state represents a “physicalproperty”. This is a strong assumption, but its content seemsclear enough. Irreducibility is usually regarded as a benignassumption, in that a reducible system can be decomposed into itsirreducible parts, to each of which Piron’s Theorem applies.

The covering law presents a more delicate problem. While it isprobably safe to say that no simple and entirely compelling argumenthas been given for assuming its general validity, Piron [1964, 1976]and others (e.g., Beltrametti and Cassinelli [1981] and Guz [1978])have derived the covering law from assumptions about the way in whichmeasurement results warrant inference from an initial state to a finalstate. Here is a brief sketch of how this argument goes. Suppose thatthere is some reasonable way to define, for an initial state \(q\) ofthe system, represented by an atom of the logic/property lattice\(L\), a final state \(\phi_p (q)\)—either another atom, orperhaps 0—conditional on the proposition \(p\) having beenconfirmed. Various arguments can be adduced suggesting that the onlyreasonable candidate for such a mapping is theSasakiprojection \(\phi_p : L \rightarrow L\), defined by

\(\phi_p (q) = (q \vee p') \wedge p\).^[18]

It can be shown that an atomic OML satisfies the atomic covering lawjust in case Sasaki projections take atoms again to atoms, or to 0.Another interesting view of the covering law is developed by Cohen andSvetlichny [1987].

6. Classical Representations

The perennial question in the interpretation of quantum mechanics isthat of whether or not essentially classical explanations areavailable, even in principle, for quantum-mechanical phenomena.Quantum logic has played a large role in shaping (and clarifying) thisdiscussion, in particular by allowing us to be quite precise aboutwhat wemean by a classical explanation.

6.1 Classical Embeddings

Suppose we are given a statistical model \((\mathcal{A},\Delta)\). Avery straightforward approach to constructing a “classicalinterpretation” of \((\mathcal{A},\Delta)\) would begin bytrying to embed \(\mathcal{A}\) in a Borel test space \(\mathcal{B}\),with the hope of then accounting for the statistical states in\(\delta\) as averages over “hidden” classical—thatis, dispersion-free—states on the latter. Thus, we’d wantto find a set \(S\) and a mapping \(X \rightarrow \wp(S)\) assigningto each outcome \(x\) of \(\mathcal{A}\) aset \(x* \subseteqS\) in such a way that, for each test \(E \in \mathcal{A}, \{x* \mid x\in E\}\) forms a partition of \(S\). If this can be done, then eachoutcome \(x\) of \(\mathcal{A}\) simply records the fact that thesystem is in one of a certain set of states, namely, \(x\)*. If we let\(\Sigma\) be the \(\sigma\)-algebra of sets generated by sets of theform \(\{x* \mid x \in X\}\), we find that each probability measure\(\mu\) on \(\Sigma\) pulls back to a state \(\mu\)* on\(\mathcal{A}\), namely, \(\mu *(x) = \mu(x\)*). So long as everystate in \(\delta\) is of this form, we may claim to have given acompletely classical interpretation of the model\((\mathcal{A},\Delta)\).

The minimal candidate for \(S\) is the set ofalldispersion-free states on \(\mathcal{A}\). Setting \(x* = \{s\in S\mid s(x) = 1\}\) gives us a classical interpretation as above, whichI’ll call theclassical image of \(\mathcal{A}\). Anyother classical interpretation factors through this one. Notice,however, that the mapping \(x \rightarrow x\)* is injective only ifthere are sufficiently many dispersion-free states to separatedistinct outcomes of \(\mathcal{A}\). If \(\mathcal{A}\) has \(no\)dispersion-free states at all, then its classical image isempty. Gleason’s theorem tells us that this is the casefor quantum-mechanical models. Thus, this particular kind of classicalexplanation is not available for quantum mechanical models.

It is sometimes overlooked that, even if a test space \(\mathcal{A}\)does have a separating set of dispersion-free states, there may existstatistical states on \(\mathcal{A}\) thatcan not berealized as mixtures of these. The classical image provides noexplanation for such states. For a very simple example of this sort ofthing, consider the test space:

and the state\(\omega(a) = \omega(b) = \omega(c) = \frac{1}{2}\), \(\omega(x) =\omega(y) = \omega(z) = 0\). It is a simple exercise to show that\(\omega\) cannot be expressed as a weighted average of\(\{0,1\}\)-valued states on \(\mathcal{A}\). For further examples anddiscussion of this point, see Wright [1980].

6.2 Contextual Hidden Variables

The upshot of the foregoing discussion is that most test spacescan’t be embedded into any classical test space, and that evenwhere such an embedding exists, it typically fails to account for someof the model’s states. However, there is one very importantclass of models for which a satisfactory classical interpretation isalways possible. Let us call a test space \(\mathcal{A}\)semi-classical if its tests do not overlap; i.e., if \(E \capF = \varnothing\) for \(E, F \in \mathcal{A}\), with \(E\ne F\).

As long as \(\mathcal{A}\) is locally countable (i.e., no test \(E\)in \(\mathcal{A}\) is uncountable), every state can be represented asa convex combination, in a suitable sense, of extreme states (Wilce[1992]). Thus, every state of a locally countable semi-classical testspace has a classical interpretation.

Even though neither Borel test spaces nor quantum test spaces aresemi-classical, one might argue that in any real laboratory situation,semi-classicality is the rule. Ordinarily, when one writes down inone’s laboratory notebook that one has performed a given testand obtained a given outcome, one always has a record of which testwas performed. Indeed, given any test space \(\mathcal{A}\), we mayalways form a semi-classical test space simply by forming theco-product (disjoint union) of the tests in \(\mathcal{A}\). Moreformally:

We can regard \(\mathcal{A}\) as arising from \(\mathcal{A}^{\sim}\)by deletion of the record of which test was performed to secure agiven outcome. Note that every state on \(\mathcal{A}\) defines astate \(\omega^{\sim}\) on \(\mathcal{A}^{\sim}\) by \(\omega^{\sim}(x,E) = \omega(x)\). The mapping \(\omega \rightarrow \omega^{\sim}\)is plainly injective; thus, we may identify the state-space of\(\mathcal{A}\) with a subset of the state-space of\(\mathcal{A}^{\sim}\). Notice that there will typically be manystates on \(\mathcal{A}^{\sim}\) thatdo not descend tostates on \(\mathcal{A}\). We might wish to think of these as“non-physical”, since they do not respect the (presumably,physically motivated) outcome-identifications whereby \(\mathcal{A}\)is defined.

Since it is semi-classical, \(\mathcal{A}^{\sim}\) admits a classicalinterpretation, as per Lemma 7.1. Let’s examine this. An elementof \(S(\mathcal{A}^{\sim}\)) amounts to a mapping \(f :\mathcal{A}^{\sim} \rightarrow X\), assigning to each test \(E \in\mathcal{A}\), an outcome \(f(E) \in E\). This is a (rather brutal)example of what is meant by acontextual (dispersion-free) hiddenvariable. The construction above tells us that such contextualhidden variables will be available for statistical models quitegenerally. For other results to the same effect, see Kochen andSpecker [1967], Gudder [1970], Holevo [1982], and, in a differentdirection, Pitowsky [1989].^[19]

Note that the simple random variables on \(\mathcal{A}\) correspondexactly to the simple random variables on \(\mathcal{A}^{\sim}\), andthat these, in turn, correspond tosome of the simple randomvariables (in the usual sense) on the measurable space\(S(\mathcal{A}^{\sim}\)). Thus, we have the following picture: Themodel \((\mathcal{A},\Delta)\) can always be obtained from a classicalmodel simply by omitting some random variables, and identifyingoutcomes that can no longer be distinguished by those that remain.

All of this might suggest that our generalized probability theorypresents no significant conceptual departure from classicalprobability theory. On the other hand, models constructed along theforegoing lines have a distinctly ad hoc character. In particular, theset of “physical” states in one of the classical (orsemi-classical) models constructed above is determined not by anyindependent physical principle, but only by consistency with theoriginal, non-semiclassical model. Another objection is that thecontextual hidden variables introduced in this section are badlynon-local. It is by now widely recognized that this non-locality isthe principal locus of non-classicality in quantum (and more general)probability models. (For more on this, see the entry onBell’s theorem.)

7. Composite Systems

Some of the most puzzling features of quantum mechanics arise inconnection with attempts to describe compound physical systems. It isin this context, for instance, that both the measurement problem andthe non-locality results centered on Bell’s theorem arise. It isinteresting that coupled systems also present a challenge to thequantum-logical programme. I will conclude this article with adescription of two results that show that the coupling ofquantum-logical models tends to move us further from the realm ofHilbert space quantum mechanics.

7.1 The Foulis-Randall Example

A particularly striking result in this connection is the observationof Foulis and Randall [1981a] that any reasonable (and reasonablygeneral) tensor product of orthoalgebras will fail to preserveortho-coherence. Consider the test space

consisting offive three-outcome tests pasted together in a loop. This test space isby no means pathological; it is both ortho-coherent and algebraic, andits logic is an orthomodular lattice. Moreover, it admits a separatingset of dispersion-free states and hence, a classical interpretation.It can also be embedded in the test space \(\mathcal{A}_{\mathbf{H}}\)of any 3-dimensional Hilbert space \(\mathbf{H}\). Now consider how wemight model a compound system consisting of two separated sub-systemseach modeled by \(\mathcal{A}_5\). We would need to construct a testspace \(\mathcal{B}\) and a mapping \(\otimes : X \times X \rightarrowY = \cup \mathcal{B}\) satisfying, minimally, the following;

1. For all outcomes \(x, y, z \in X\), if \(x\binbot y\), then\(x\otimes z \binbot y\otimes z\) and \(z\otimes x \binbot z\otimesy\),
2. For each pair of states \(\alpha , \beta \in\Omega(\mathcal{A}_5)\), there exists at least one state \(\omega\) on\(\mathcal{B}\) such that \(\omega(x\otimes y) = \alpha(x)\beta(y)\),for all outcomes \(x, y \in X\).

Foulis and Randall show that no such embedding exists for which\(\mathcal{B}\) is orthocoherent. Indeed, suppose we have a test space\(\mathcal{B}\) and an embedding satisfying conditions (a) and (b).Consider the set of outcomes

By (a), this set is pairwiseorthogonal. Now let \(\alpha\) be the state on \(\mathcal{A}_5\)taking the value 1/2 on outcomes \(a, b, c, d\) and \(e\), and thevalue 0 on \(x, y, z, w\) and \(v\). By condition (b), there existsstate \(\omega\) on \(\mathcal{B}\) such that

for alloutcomes \(s, t\) in \(X\). But this state takes the constant value1/4 on the set \(S\), whence, it sums over this set to \(5/4 \gt 1\).Hence, \(S\) is not an event, and \(\mathcal{B}\) is notorthocoherent.

It is important to emphasize here that the test space\(\mathcal{A}_5\) has a perfectly unproblematic quantum-mechanicalinterpretation, as it can be realized as a set of orthonormal bases ina 3-dimensional Hilbert space \(\mathbf{H}\). However, thestate \(\omega\) figuring in the Foulis-Randall examplecannot arise quantum-mechanically (much less classically). (Indeed,this follows from the example itself: the canonical mapping\(\mathbf{H} \times \mathbf{H} \rightarrow \mathbf{H} \otimes\mathbf{H}\) provides a mapping satisfying the conditions (a) and (b)above. Since \(\mathbf{L}(\mathbf{H} \otimes \mathbf{H})\) isorthocoherent, the set S corresponds to a pairwise orthogonal familyof projections, over which a quantum-mechanical state would have tosum to no more than 1.)

7.2 Aerts’ Theorem

Another result having a somewhat similar force is that of Aerts[1981]. If \(L_1\) and \(L_2\) are two Piron lattices, Aertsconstructs in a rather natural way a lattice \(L\) representing twoseparated systems, each modeled by one of the given lattices.Here “separated” means that each pure state of the largersystem \(L\) is entirely determined by the states of the two componentsystems \(L_1\) and \(L_2\). Aerts then shows that \(L\) is again aPiron lattice iff at least one of the two factors \(L_1\) and \(L_2\)is classical. (This result has recently been strengthened by Ischi[2000] in several ways.)

7.3 Ramifications

The thrust of these no-go results is that straightforwardconstructions of plausible models for composite systems destroyregularity conditions (ortho-coherence in the case of theFoulis-Randall result, orthomodularity and the covering law in that ofAerts’ result) that have widely been used to underwritereconstructions of the usual quantum-mechanical formalism. This putsin doubt whether any of these conditions can be regarded as having theuniversality that the most optimistic version of Mackey’sprogramme asks for. Of course, this does not rule out the possibilitythat these conditions may yet be motivated in the case of especiallysimple physical systems.

In some quarters, the fact that the most traditional models of quantumlogics lack a reasonable tensor product have have been seen asheralding the collapse of the entire quantum-logical enterprise. Thisreaction is premature. The Foulis-Randall example, for instance, showsthat there can be no general tensor product that behaves properly onall orthomodular lattices or orthomodular posets (that is,orthocoherent orthoalgebras),and onall statesthereon. But this does not rule out the existence of a satisfactory tensorproduct for classes of structureslarger than that oforthomodular posets, orsmaller than that of orthomodularlattices, or for classes of orthomodular lattices or posets withrestricted state spaces. Quantum Mechanics itself provides one example. For another, as Foulis and Randall showed inFoulis and Randall [1981a], the class of unitalorthoalgebras—that is, orthoalgebras in which every propositionhas probability 1 in some state—does support acanonical tensor product satisfying their conditions (a) and (b).

Moving in the opposite direction, one can take it as an axiomaticrequirement that a satisfactory physical theory be closed under somereasonable device for coupling separated systems. This suggests takingclasses of systems, i.e., physical theories, as distinct fromindividual systems, as the focus of attention. And in fact, this isexactly the trend in much current work on the foundations of quantummechanics.

A particularly fruitful approach of this kind, due to Abramsky andCoecke [2009] takes a physical theory to be represented by a symmetricmonoidal category—roughly, a category equipped with a naturallysymmetric and associative tensor product. Subject to some furtherconstraints (e.g., compact closure), such categories exhibit formalproperties strikingly reminiscent of quantum mechanics. Interestingly,it has recently been shown by Harding [2009] that, in every stronglycompact closed category with biproducts, every object is associatedwith an orthomodular poset Proj\((A)\) of “weakprojections”, and that Proj\((A \otimes B)\) behaves in manyrespects as a sensible tensor product for Proj\((A)\) and Proj\((B)\).From this perspective, the FR example simply exhibits a pathologicalexample — \(A_5\) and the state \(\alpha\) — that can notbe accommodated in such a theory, establishing that the monoidalityrequirement imposes a nontrivial restriction on the structure ofindividual systems.

This recent emphasis on systems in interaction is part of a moregeneral shift of attention away from the static structure of statesand observables and towards theprocesses in which physicalsystems can participate. This trend is evident not only in thecategory-theoretic formulation of Abramsky and Coecke (see also Coecke[2011]), but also in several recent axiomatic reconstructions ofquantum theory (e.g., Hardy [2001, Other Internet Resources], Rau[2009], Dakic-Brukner [2011], Massanes and Mueller [2011],Chiribella-D’Ariano-Perinotti [2011], Wilce [2018]), most ofwhich involve assumptions about how physical systems combine. In adifferent direction, Baltag and Smets [2005] enrich a Piron-stylelattice-theoretic framework with an explicitly dynamical element,arriving at a quantum analogue of propositional dynamical logic.

8. Effect Algebras

Another recent development was the introduction in the early 1990s ofstructures calledeffect algebras (Foulis and Bennett[1994]) generalizing the orthoalgebras discussed in sect 4.1. Thedefinition is almost identical, except that the weaker condition \(a\perp a \Rightarrow a = 0\) is replaced by the weaker condition \(a\perp 1 \ \Rightarrow \ a = 0\). Like orthoalgebras, effect algebrasare partially ordered by setting \(a \leq b\) iff \(b = a \oplus c\)for some \(c \perp a\).^[20]

A simple but important example is the effect algebra \([0,1]^{E}\) offunctions \(\,f : E \rightarrow [0,1]\), with \(f \perp g\) iff \(f + g\leq 1\) and, in that case, \(f \oplus g = f + g\). One can regardelements of \([0,1]^{E}\) as “unsharp” or“fuzzy” versions of indicator functions \(f : E\rightarrow \{0,1\}\). The set \(\{0,1\}^{E}\) of indicatorfunctions, regarded as a subeffect algebra of \([0,1]^{E}\), is anorthoalgebra and, of course, isomorphic to the boolean algebra ofsubsets of \(E\).^[21]

Effect algebras exist in great abundance. In particular, if \(\Omega\)is a convex set arising as the state-space of a probabilistic model,then the set \({\mathcal E}(\Omega)\) of bounded affine(convex-linear) functions \(f : \Omega \rightarrow [0,1]\) form aneffect algebra, with \(f \oplus g = f + g\) if \(f + g \leq 1\). Theidea is that a function \(\,f \in {\mathcal E}(\Omega)\) represents an"in principle" measurement outcome, with probability \(f(\alpha)\) instate \(\alpha \in \Omega\). If \(f_0,...,f_n \in {\mathcalE}(\Omega)\) with \(f_0 + \cdots + f_n = 1\), then the sequence\((f_0,...,f_n)\) rpresents an “in principle” observable with values\(i = 0,...,n\), taking value \(i\) with probability\(f_i(\alpha)\).

8.1 Quantum Effects and Naimark’s Theorem

In the special case where \(\Omega = \Omega(\mathbf{H})\), the set ofdensity operators on a Hilbert space \(\mathbf{H}\), one can showthat every effect \(f\) on \(\Omega\) has the form \(\,f(W) =\textrm{Tr}(W a)\) for a unique positive self-adjoint operator \(a\)with \(a \leq 1\). Conversely, such an operatordefines an effect through the formula just given. One thereforeidentifies \(\mathcal{E}(\Omega(\mathbf{H}))\) with the set\(\mathcal{E}(\mathbf{H})\) of all positive self-adjoint operators on\(\mathbf{H})\) with \(0 \leq a \leq 1\), referring to these also aseffects.

Arbitrary quantum effects, and arbitrary effect-valued observables,arise quite naturally as models of actual experimentaloutcomes. Consider an isolated quantum system \(A\) with Hilbert space\(\mathbf{H}_A\), and an ancillary system \(B\), with Hilbert space\(\mathbf{H}_{B}\), maintained in a reference state represented by adensity operator \(W^{B}_o\). If \(A\) is in the state represented bya density operator \(W^{A}\) on \(\mathbf{H}_A\), thet state of thejoint system is represented by \(W^{A} \otimes W^{B}_o\). If we make ayes-no measurement on \(AB\) represented by a projection operator\(P_{AB}\) on \(\mathbf{H}_{AB} = \mathbf{H}_{A} \otimes\mathbf{H}_{B}\) then the probability of obtaining a positive resultis \(\textrm{Tr}(P_{AB}(W^{A} \otimes W^{B}_{o}))\). This defines abounded convex-linear function of \(W^{A}\), and hence, there is aunique effect \(a\) with \(\textrm{Tr}((W^{A} \otimesW^{B}_{o})P_{AB}) = \textrm{Tr}(W^{A} a)\). This effect \(a\) iscalled thecompression of \(P_{AB}\) onto \(\mathbf{H}_{A}.\)In other words, we can understand \(a\) as representing the result ofmeasuring \(P_{AB}\) on the combined system \(AB\), holding \(B\) instate \(W^{B}_o\), and then “forgetting about” theancillary system \(B\).It is not difficult to show that every every effect on \(A\) arises inthis way from a projection on \(\mathbf{H}_{A} \otimes\mathbf{H}_{B}\) for a suitable Hilbert space \(\mathbf{H}_{B}\). Moregenerally, a classic result in operator theory known asNaimark’sTheorem asserts that any effect-valued observable\(a_1,...,a_n\) on \(A\) arises by compression of an ordinaryprojection-valued observable \(P_1,...,P_n\) on \(AB\) for a suitablequantum system \(B\). Thus, all effects, and indeed all effect-valuedobservables, on \(A\) are physically realizable. In view of this, itis difficult to see why effect algebras should have any less claim tothe status of a “quantum logic” than do, say, orthomodular posets.

8.2 Sequential effect algebras

A natural question is whether one can characterize those effectalgebras of the special form \(\mathcal{E}(\mathbf{H})\). One way inwhich effects arise naturally is in the context of sequentialmeasurements. If \(P\) is a projection, a measurement of \(P\) instate corresponding to the density operator \(W\) leaves the system inthe state corresponding to the density operator

A subsequent measurement of \(q\) in thisstate then yields a positive result with probability \begin{equation}\textrm{Tr}(W_{P} Q) = \frac{\textrm{Tr}(QP W PQ)}{\textrm{Tr}(W P)} =\frac{\textrm{Tr}(W PQP)}{\textrm{Tr}(W P)}. \end{equation} Theoperator \(PQP\) is not a projection unless \(P\) and \(Q\) commute,but is always an effect. If we write \(\Pr(a|W)\) for\(\textrm{Tr}(Wa)\) for arbitrary effects \(a\), then the above can berewritten, perhaps more transparently, as

Thus, \(PQP\) represents the “(yes,yes)”-outcome in asequential measurement of \(P\) and \(Q\) (in that order).

More generally, the sequential product \(a \odot b :=\sqrt{a}b\sqrt{a}\) of two effects is another effect, representing theresult of observing first \(a\) and then \(b\) in a sequentialmeasurement (and assuming the state updates according to \(W \mapsto(\textrm{Tr}(Wa))^{-1} \sqrt{a} W \sqrt{a}\) after measurement of\(a\)). Abstracting from this example, S. Gudder and R. J. Greechie([2002]) defined asequential effect algebra to be an effectalgebra \((\mathbf{L},\oplus,0,1)\) equipped with a binary operation\(\odot : \mathbf{L} \times \mathbf{L} \rightarrow \mathbf{L}\)satisfying the following conditions for all \(a,b,c \in \mathbf{L}\),where \(a | b\) means \(a \odot b = b \odot a\):

• If \(b \perp c\), then \(a \circ b \perp a \circ c\) and \(a \circ (b \oplus c) = (a \circ b) \oplus (a \circ c)\)
• \(1 \circ a = a\)
• If \(a \circ b = 0\), then \(b \circ a = 0\)
• If \(a | b\) then \(a | b'\), and \(a \circ (b \circ c) = (a \circb) \circ c\)
• If \(a | b\) and \(a | c\), then (i) \(a | b \circ c\) and, (ii)if \(b \perp c\), \(c | a \oplus b\)

A remarkable recent result of J. van de Wetering ([2019]) shows thatany finite-dimensional order-unit space whose order interval \([0,u]\)is an SEA under a binary operation continuous in the first variable,is a euclidean (equivalently, formally real) Jordan algebra in anatural way.^[22]

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• Cabello, Adan, 2012, “Specker’s fundamental principle of quantum mechanics”, unpublished paper, available at arXiv.org
• Hardy, Lucien, 2001, “Quantum Theory from Five Reasonable Axioms”, unpublished paper, available at arXiv.org
• Henson, Joe, 2012, “Quantum contextuality from a simple principle?”, unpublished manuscript, available at arXiv.org

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