arX

iv:q

uant

-ph/

0309

017v

1 1

Sep

200

3

Noncontextuality, Finite Precision Measurement

and the Kochen-Specker Theorem

Jonathan Barrett

Service de Physique Theorique

Universite Libre de Bruxelles,

CP 225, Bvd. du Triomphe

1050 Bruxelles Belgium

Adrian Kent

Centre for Quantum Computation

Department of Applied Mathematics and Theoretical Physics

University of Cambridge

Wilberforce Road, Cambridge CB3 0WA

United Kingdom

August 2003

Abstract

Meyer originally raised the question of whether non-contextual hid-

den variable models can, despite the Kochen-Specker theorem, simulate

the predictions of quantum mechanics to within any fixed finite experi-

mental precision. Clifton and Kent later presented constructions of non-

contextual hidden variable theories which, they argued, indeed simulate

quantum mechanics in this way.

These arguments have evoked some controversy. Among other things,

it has been suggested that the CK models do not in fact reproduce cor-

rectly the predictions of quantum mechanics, even when finite precision

is taken into account. It has also been suggested that careful analysis

of the notion of contextuality in the context of finite precision measure-

ment motivates definitions which imply that the CK models are in fact

contextual. Several critics have also argued that the issue can be defini-

tively resolved by experimental tests of the Kochen-Specker theorem or

experimental demonstrations of the contextuality of Nature.

One aim of this paper is to respond to and rebut criticisms of the MCK

papers. We thus elaborate in a little more detail how the CK models can

reproduce the predictions of quantum mechanics to arbitrary precision.

We analyse in more detail the relationship between classicality, finite pre-

cision measurement and contextuality, and defend the claims that the CK

models are both essentially classical and non-contextual. We also examine

1

in more detail the senses in which a theory can be said to be contextual

or non-contextual, and in which an experiment can be said to provide

evidence on the point. In particular, we criticise the suggestion that a de-

cisive experimental verification of contextuality is possible, arguing that

the idea rests on a conceptual confusion.

1 Introduction

1.1 The Kochen-Specker theorem

Consider a set K of Hermitian operators that act on an n-dimensional Hilbertspace. Suppose that V is a map that takes a Hermitian operator in K to a realnumber in its spectrum. We call such a map a colouring of K. If the followingconditions are satisfied

V (A+ B) = V (A) + V (B)

V (AB) = V (A)V (B)

∀A, B ∈ K s.t.[A, B] = 0, (1)

then the map is a KS-colouring of K. We call these conditions the KS criteria.Kochen and Specker’s celebrated theorem[1, 2] states that if n > 2 there areKS-uncolourable sets, i.e. sets K for which no KS-colouring exists. It followstrivially that the set of all Hermitian operators acting on a Hilbert space ofdimension > 2 is KS-uncolourable.

As Bell [3] first pointed out, this result is a corollary of Gleason’s theorem.Kochen and Specker provided an independent proof. Many proofs along thelines of Kochen and Specker’s have since been produced (e.g. [4, 5, 6, 7]) byconstructing demonstrably KS-uncolourable sets. The most common type ofproof describes a set of 1-dimensional projection operators in n dimensions thatis KS-uncolourable. If we represent 1-dimensional projections by vectors ontowhich they project, and colour the corresponding set of vectors with a 1 or a 0,the KS criteria would imply that for each orthogonal n-tuple of vectors, exactlyone must be coloured 1, and all the rest 0. The Kochen-Specker theorem canthen be proved by showing that the colouring condition cannot be satisfied.In their original proof, Kochen and Specker describe a set of 117 vectors in 3dimensions that is KS-uncolourable.1

Of course, the well-known proofs[2, 3, 4, 5, 6, 7] of the Kochen-Specker the-orem are logically correct. Moreover, the Kochen-Specker theorem undeniablysays something very important and interesting about fundamental physics: itshows that the predictions of quantum theory for the outcomes of measurementsof hermitian operators belonging to a KS-uncolourable set cannot be preciselybe reproduced by any hidden variable theory which assigns real values to theseoperators in a way that respects the KS criteria, since no such hidden variable

1How small can a KS-uncolourable set of vectors be? The current records stand at 31vectors in 3 dimensions [6] and 18 in 4 dimensions[7].

2

theory exists. However, debate continues over the extent to which Kochen andSpecker succeeded in their stated aim, “to give a proof of the nonexistence ofhidden variables”([2], p.59) even when this is qualified (as it must be) by re-stricting attention to noncontextual hidden variables. Before summarising andcontinuing this debate, we review why one might be interested in hidden variabletheories in the first place.

Consider a system in a state |ψ〉 and a set of observables A,B,C, . . . suchthat |ψ〉 is not an eigenstate of A, B, C, . . .; here we use capital letters withhats to denote Hermitian operators and capital letters without hats to denotethe corresponding observables. Orthodox quantum mechanics leads us to saysomething like this: if we measure A, we will obtain the result a with probabilitypa, if we measure B, we will obtain the result b with probability pb, and so on.With an ease born of familiarity, the well trained quantum mechanic will notbat an eyelid at such statements. But, one might well ask: why are they sooddly phrased? Could this just be a rather awkward way of saying that withprobability pa, the value of A is a, or with probability pb, the value of B is b,and so on?

Suppose that the set A,B,C, . . . corresponds to a KS-uncolourable set ofoperators A, B, C, . . .. The suggestion is that at a given time, each observablein the set has some definite value associated with it, defined by some “hidden”variables of the system. The significance of the KS criteria is that if the Her-mitian operators associated with two observables commute, then according toquantum mechanics, the observables can be simultaneously measured, and thevalues obtained will satisfy the KS criteria (and in general will satisfy any func-tional relationships that hold between the operators themselves). We are notlogically compelled to assume that any hidden variable theory shares these prop-erties. However, the standard motivation for considering hidden variables is toexamine the possibility that quantum theory, while not incorrect, is incomplete,Thus motivated, it seems natural to assume that the colouring defined by thehidden variables must also satisfy the KS criteria, But given this assumption,since there is no such colouring, the original supposition that the observableshave definite values must be wrong.

The contradiction obtained in the Kochen-Specker theorem is avoided if,instead of defining a map V , we assign values to Hermitian operators in sucha way that the value assigned to a particular Hermitian operator depends onwhich commuting set we are considering that operator to be part of. Such avalue assignment is called contextual. Hidden variable interpretations of quan-tum theory based on contextual value assignments can be defined. In suchcontextual hidden variables (CHV) interpretations, the outcome obtained onmeasuring a certain quantum mechanical observable is indeed pre-defined, butdepends in general on which other quantum mechanical observables are mea-sured at the same time. Thus, if we take the KS criteria for granted, Kochenand Specker’s results show that there are no non-contextual hidden variables

(NCHV) interpretations of the standard quantum mechanical formalism.It may seem tempting to phrase this more directly, concluding that the

Kochen- Specker theorem shows that Nature cannot be described by any non-

3

contextual hidden variable theory. Another possible conclusion is that the KStheorem implies that we could exclude non-contextual hidden variable theoriesif the predictions of quantum theory were confirmed in a suitably designedexperiment. We will argue below that neither conclusion is correct.

1.2 Querying the scope of the KS theorem

We next review some earlier discussions that suggest limitations on what canbe inferred from the Kochen-Specker theorem.

Some time ago, Pitowsky[9, 10] devised models which assign values non-contextually to the orthogonal projections in three dimensions and nonethelesssatisfy (1) “almost everywhere”[10]. The models are non-constructive, requiringthe axiom of choice and the continuum hypothesis (or some suitable weakerassumption) for their definition. Another complication is that the term “almosteverywhere” is not meant in the standard sense, but with respect to a non-standard version of measure theory proposed by Pitowsky[9] which, among otherdisconcerting features, allows the intersection of two sets of probability measure1 to have probability measure 0.[8]

Pitowsky’s models disagree with quantum mechanics for some measurementchoices, as the KS theorem shows they must. They thus do not per se seemto pose an insuperable obstacle to arguments which — either directly from thetheorem or with the aid of suitable experiments — purport to demonstrate thecontextuality of Nature.2 After all, either the demonstration of a finite non-colourable set of projectors is sufficient to run an argument, or it is not. If it is,Pitowsky’s models are irrelevant to the point; if it is not, it is not obvious thatthe models. equipped as they are with an entirely novel version of probabilitytheory, are either necessary or sufficient for a refutation.

A more direct challenge to the possibility of theoretical or experimentalrefutations of non-contextual hidden variables was presented by Meyer[12], whoemphasized the implications of finite experimental precision: “Only finite preci-sion measurements are experimentally reasonable, and they cannot distinguisha dense subset from its closure.”[12] Meyer identified a particularly simple andelegant construction, originally due to Godsil and Zaks[13], of a KS-colourabledense subset of the set of projectors in three dimensions.3 His conclusion wasthat, at least in three dimensions, the Kochen-Specker theorem could be “nul-lified”.4 As a corollary, Meyer argued that the KS theorem alone cannot dis-criminate between quantum and classical (therefore non-contextual) informationprocessing systems.

Meyer left open the question of whether static non-contextual hidden vari-able theories reproducing the predictions of quantum theory for three dimen-

2Nor, it should be stressed, did Pitowsky suggest that they do.3As Pitowsky has since noted, Meyer’s argument could also be framed using one of

Pitowsky’s constructions of dense KS-colourable sets of projectors rather than Godsil andZaks’.

4It should perhaps be emphasized that the sense of “nullify” intended here is “counteractthe force or effectiveness of”, not “invalidate”. Neither Meyer nor anyone else has suggestedthat the proofs of the Kochen-Specker theorem are flawed.

4

sional systems actually exist: his point was that, contrary to most previousexpectations, the Kochen-Specker theorem does not preclude such hidden vari-able theories.

Meyer’s result was subsequently extended by a construction of KS-colourabledense sets of projectors in complex Hilbert spaces of arbitrary dimension [14].Clifton and Kent (CK) extended the result further [15] by demonstrating theexistence of dense sets of projection operators, in complex Hilbert spaces ofarbitrary dimension, with the property that no two compatible projectors aremembers of incompatible resolutions of the identity. The significance of thisproperty is that it makes it trivial to construct a distribution over different hid-den states that recovers the quantum mechanical expectation values. Such a dis-tribution is, of course, necessary for a static hidden variable theory to reproducethe predictions of quantum theory. Similar constructions of dense subsets of thesets of all positive operators were also demonstrated[14, 15]. CK presented[15]their constructions as non-contextual hidden variable theories which can indeedsimulate the predictions of quantum mechanics in the sense that the theoriesare indistinguishable in real experiments in which the measurement operatorsare defined with finite precision.

The arguments set out by Meyer, Kent and Clifton (MKC) have evoked somecontroversy (see e.g. Refs. [16, 17, 21, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33]).Among other things, it has been suggested[32] that the models do not in factreproduce correctly the predictions of quantum mechanics, even when finiteprecision is taken into account. It has also been suggested[24, 25, 27] thatcareful analysis of the notion of contextuality in the context of finite precisionmeasurement motivates definitions which imply that the CK models are in factcontextual.

Several of these critiques raise novel and interesting points, which have ad-vanced our understanding of the Kochen-Specker theorem and its implications.Nonetheless, we remain convinced that Meyer’s essential insight[12] and all thesubstantial points made in Refs. [14, 15] are valid. One aim of this paper isthus to respond to and rebut MKC’s critics.

Perhaps unsurprisingly, quite a few critics have made similar points. Also,some purportedly critical arguments make points irrelevant to the argumentsof the MKC papers (which were carefully limited in their scope). Rather thanproducing a comprehensive — but, we fear, unreadable — collection of counter-critiques of each critical article, we have tried in this paper to summarize andcomment on the most interesting new lines of argument.

Among other things, we explain here in a little more detail how the MKCmodels can reproduce the predictions of quantum mechanics to arbitrary preci-sion, both for single measurements and for sequences. We point out a conceptualconfusion among critics who suggest that the models are contextual, notingthat the arguments used would (incorrectly) suggest that Newtonian physicsand other classical theories are contextual. We also defend the claim that theCK models are essentially classical. Indeed, as we explain, the models showin principle that one can construct classical devices that assign measurementoutcomes non-contextually and yet simulate quantum mechanics to any given

5

fixed nonzero precision. In summary, we reiterate the original claim of MKCthat the models, via finite precision, provide a loophole — which is physicallyimplausible but logically possible — in the Kochen-Specker argument.

Running through these debates is another theme: the alleged possibility ofexperimental tests of the Kochen-Specker theorem, or experimental demonstra-tions of the contextuality of Nature. Such experiments, it has been suggested,could be analogous to the Bell-type tests of local causality. Several such exper-iments have been proposed, some before the MKC results, without allowing forfinite experimental precision[18, 19], and some after, attempting to take finiteprecision into account [21, 24, 25]. Another aim of this paper is to go beyondprevious discussions in examining in detail the senses in which a theory can besaid to be contextual or non-contextual, and in which an experiment can be saidto provide evidence for these. Broadly, we are critical of the idea of an exper-imental test of non-contextuality. We argue that the idea rests on conceptualconfusion and that none of the proposed experiments, if actually performed,would be of decisive significance.

We believe that careful consideration of these various points should dispelany lingering controversies.

2 Experimental tests of contextuality?

We begin with a discussion of some earlier attempts at describing experimentaltests of the Kochen-Specker theorem. Cabello and Garcia-Alcaine (CG) [18] firstdescribed such a test from a theoretical point of view. Simon et al. (SZWZ)[19] then described a realisable laboratory implementation of a similar scheme.These papers were published before the MKC papers and do not consider theimplications of finite experimental precision. Obviously, they can be criticised onthis score. Our aim in this section, though, is to examine the authors’ accountsof their experiments, granting for the moment the unrealistic assumption thatperfect experimental precision could be obtained. The questions raised go to theheart of our understanding of the conceptual implications of the Kochen-Speckertheorem and the very possibility of an experimental test, and have bearing onthe discussion of the MKC models that follows.

The essential points of our discussion apply to both proposals; for defi-niteness, we focus here on SZWZ’s. SZWZ consider a 4-dimensional Hilbertspace which we can think of as representing two 2-dimensional subsystems.(In their actual example, the two subsystems are associated with the spin andpath degrees of freedom of a single particle.) Define the subsystem observablesZ1, X1, Z2, X2, where the subscript indicates which subsystem the observableis associated with. Suppose that Zi = σzi and Xi = σxi, where σzi and σxiare Pauli operators acting on subsystem i. Each of these observables can takethe values +1,−1. In an NCHV interpretation, a hidden state must assign avalue to each of these observables that would simply be revealed on measure-ment. This in turn defines a colouring of the corresponding set of operators,V (Z1), V (X1), V (Z2), V (X2).

6

One can also consider observables that are products of these observables, forexample, Z1X2. Product observables also take the values +1,−1, and from theKS criteria we have:

V (Z1Z2) = V (Z1)V (Z2)

V (Z1X2) = V (Z1)V (X2)

V (X1Z2) = V (X1)V (Z2)

V (X1X2) = V (X1)V (X2) . (2)

As noted in the last section, one argues for this last claim by noting that oneway of measuring a product observable like Z1Z2 is simply to measure theobservables Z1 and Z2 separately and to multiply the results. Finally, thecontradiction arises on consideration of the quantum state

|φ+〉 =1√2(|+ z〉|+ z〉+ | − z〉| − z〉)

=1√2(|+ x〉|+ x〉 + | − x〉| − x〉),

where |+ z〉 is an eigenstate of Zi with eigenvalue +1, and so on. This state hasthe property that measurement of the product Z1Z2 always returns 1, as doesmeasurement of X1X2. If V (Z1) = V (Z2), V (X1) = V (X2), and Eqs. (2) aresatisfied, then it follows logically that V (Z1X2) = V (X1Z2). Yet in quantummechanics, one can measure Z1X2 and X1Z2 simultaneously, and if the stateis |φ+〉, then one will get opposite results with certainty. Hence we have acontradiction.5

SZWZ describe a laboratory implementation with beam splitters and Stern-Gerlach devices, for each of the joint measurements

(Z1, Z2), (Z1, X2), (X1, Z2), (X1, X2), (Z1X2, X1Z2) .

What might such a laboratory implementation actually show? SZWZ and CGmotivate their work via an analogy with Bell’s theorem. Bell’s theorem tells usthat locally causal theories are incompatible with quantum mechanics, accordingto Bell’s precise definition[20] of “locally causal”. The associated experimentaltests have strongly confirmed quantum mechanics. According to SZWZ,

“The Kochen-Specker theorem states that non-contextual theo-ries are incompatible with quantum mechanics.”([19], p. 1783)

If one takes this at face value, it seems easy to accept that a Kochen-Specker ex-periment to test non-contextuality would be of similar interest and fundamentalimportance to a Bell experiment that tests local causality.

5Note that this argument differs from standard Kochen-Specker-style proofs in that thepredictions from a particular quantum state are used to obtain a contradiction. Cabello’s andGarcia-Alcaine’s argument does not have this feature.

7

However, there is a key point, not noted by these authors, where the analogybreaks down. A Bell experiment allows us to test the predictions of quantummechanics against those of locally causal theories because a definition of all theterms used in a derivation of Bell’s theorem (in particular the term “locallycausal” itself) can be given that is theory independent. Yet in the Kochen-Specker scheme above, the observables have not been defined in a manner that istheory independent, but have instead been defined with respect to the quantummechanical operators. When a simultaneous measurement of Z1X2 and X1Z2 isperformed, the experimental setup looks quite different from that employed ina simultaneous measurement of, say, Z1 and Z2. What gives us license to claimthat in the former case, we really are measuring two observables, where one is theproduct of Z1 and X2 and the other is the product of X1 and Z2? The answeris: our conventional physical understanding of the experiment, as informed by

the quantum formalism. In other words, there is no theory independent meansof knowing that we really are doing a simultaneous measurement of the productof Z1 and X2, and the product of X1 and Z2. But this is crucial if we are toconclude unequivocally that contextuality is being exhibited.

Of course, the mathematical part of the argument is valid as a (further) proofthat there are no NCHV interpretations of the quantum mechanical formalism.However, to claim that this type of experiment can show that

“NCHV theories, without any call to the formal structure of QM,make conflicting predictions with those of QM” (Ref. [18]; theiremphasis)

is not correct.We conclude that experiments along the lines of those of Refs. [18, 19] do not

and cannot decisively distinguish between contextuality and non-contextualityin Nature. They need to assume the quantum formalism of states and operators.But we know already, without carrying out any experiments, that the standardquantum formalism cannot be reproduced by hidden variables assigning valuesnon-contextually to the set of all hermitian operators. Mermin’s comment that

“the whole notion of an experimental test of [the Kochen-Speckertheorem] misses the point”[22]

still seems to us to apply, at least to SZWZ and CG’s experiments.If experiments of this kind have any significance, it must be as a test be-

tween quantum mechanics and non-contextual theories of a rather restrictedkind. Such an experiment could serve as a test between quantum mechanicsand a non-contextual theory that accepts at least part of Hilbert space struc-ture (the experimenters must agree on which operators are being measured),but rejects, for example, the KS criteria. Logically, this would be a valid experi-ment. However, in order to motivate it, one would need to devise an interestingand plausible alternative to quantum theory in which (1) is violated. Consid-ering such alternatives is beyond our scope here; we only wish to note that theclass of such alternatives is not nearly as general and natural as the class of

8

locally causal theories. So far as the project of verifying the contextuality ofNature (as opposed to the contextuality of hidden variable interpretations ofthe standard quantum formalism) is concerned, the question is of rather limitedrelevance and interest.

Suppose, anyway, that such a Kochen-Specker experiment is performed, andthe quantum predictions verified. We then have three choices. First, accept thebasic quantum formalism and accept also that any underlying hidden variabletheory assigning values to hermitian operators must be contextual. Second,look for loopholes in our interpretation of the experimental results. Or, third,reject the Hilbert space structure and look for an entirely different theory of theexperiment that is non-contextual in its own terms.

The second move is exploited by the MKC models, as we discuss below.The third move will always be logically possible if non-contextuality is defined(as it often is in the literature) as simply requiring that the value obtainedon measuring a given observable does not depend on which other observablesare measured at the same time. No mention of Hermitian operators is givenin this definition, so it has the appearance of being theory independent. Butit is not all that useful. It allows a non-contextual theory of any experimentto be cooked up in a trivial manner, simply by redefining what counts as anobservable — for instance, by taking an observable to correspond to the fullprojective decomposition of the identity defining any given measurement, ratherthan to a single projection[23].

Note that if a Bell experiment is performed, and the quantum predictionsverified, then we have analogues of the first two options above: we can rejectlocal causality, or we can look for loopholes in the experiment. Both options havebeen much explored. But the analogy breaks down when we consider the thirdoption above, because the Hilbert space structure was not used either in thederivation of Bell’s theorem or in the interpretation of the experiment.6 It alsobreaks down when we consider the outcomes of exploring the second option:finite experimental precision poses no fundamental difficulty in the analysisof Bell experiments, but turns out to be an unstoppable loophole in Kochen-Specker experiments.

Finally, we note that one might try to define “observable”, and thence “non-contextual”, operationally, in the hope of giving these terms a meaning that isgenuinely theory independent. Some authors[24, 25] have recently taken thisapproach, and proposed Kochen-Specker-like inequalities, by means of which,they argue, one can show that a black box which carries out quantum exper-iments is behaving in a contextual manner. We come back to this in Sec. 5.First we introduce the MKC models and explain how finite precision will pro-

6Of course, even local causality cannot be defined with no assumptions about an underlyingtheory. It requires the notion of a background space-time with a causal structure. Bell’sdiscussion of the implication of local causality for Bell experiments also implicitly requiresthat the notion of an experimental outcome has its conventional meaning.

It is worth noting, incidentally, that this last point leaves room for arguing that an Ev-erettian interpretation of quantum theory might be defined so as to be locally causal. We willnot pursue this here, since the larger questions of whether a coherent Everettian interpretationexists, and if so on what assumptions, are beyond our present scope.

9

vide a loophole in our interpretation of any Kochen-Specker-type experiment.In Sec. 4, we discuss the models further, countering some of the more immediateobjections.

3 MKC models

Kochen and Specker’s[2] declared motivation for constructing finite uncolourablesets is intriguing, both because it partly anticipates the point made a third of acentury later by Meyer and because its implications seem to have been largelyignored in the period intervening:

It seems to us important in the demonstration of the non-existenceof hidden variables that we deal with a small finite partial Booleanalgebra. For otherwise a reasonable objection can be raised thatin fact it is not physically meaningful to assume that there are acontinuum number of quantum mechanical propositions. ([2], p.70)

What Kochen and Specker neglected to consider is that the objection mightbe sharpened: it could be that in fact only a specified countable set of quan-tum mechanical propositions exist, and it could be that this set has no KS-uncolourable subsets (finite or otherwise). This is the possibility that the MKCmodels exploit.

Before discussing these models, we wish to reemphasize the disclaimers madein Ref. [15]. MKC models describe a type of hidden variable theory that is alogically possible alternative to standard quantum theory, but not, in our view, avery plausible one. The CK constructions in particular, are ugly and contrivedmodels, produced merely to make a logical point. One might hope to deviseprettier hidden variable models which do the same job, using a colouring schemeas natural and elegant as Godsil-Zaks’. Even if such models were devised,though, we would not be inclined to take them too seriously as scientific theories.

However, we think it important to distinguish between scientific implausi-bility and logical impossibility. The models show that only the former preventsus from adopting a non-contextual interpretation of any real physical experi-ment. Another reason for studying the models — in fact, Meyer’s main originalmotivation[12] — is to glean insights into the possible role of contextuality inquantum information theory.

3.1 Projective measurements

Kochen and Specker’s argument[2], and most later discussions until recently,including Meyer’s[12], assume that the quantum theory of measurement can beframed entirely in terms of projective measurements. This remains a tenableview, so long as one is willing to accept that the experimental configuration

10

defines the quantum system being measured.7 We adopt it here, postponingdiscussion of positive operator valued measurements to the next subsection.

Meyer identified a KS-colouring, originally due to Godsil and Zaks[13], ofthe set S2 ∩Q3 of unit vectors in R3 with rational components, or equivalentlyof the projectors onto these vectors. As he pointed out, not only is this set ofprojectors dense in the set of all projectors in R3, but the corresponding set ofprojective decompositions of the identity is dense in the space of all projectivedecompositions of the identity.

Meyer’s result is enough to show that an NCHV theory along these lines isnot ruled out by the Kochen-Specker theorem. It does not show that such atheory exists. For this we need there to be KS-colourable dense sets of projec-tors in complex Hilbert spaces of arbitrary dimension. Further, it is not enoughfor each set to admit at least one KS-colouring. For each quantum state, onemust be able to define a distribution over different KS-colourings such that thecorrect quantum expectation values are obtained. For these reasons, Kent ex-tended Meyer’s result by constructing KS-colourable dense sets of projectors incomplex Hilbert spaces of arbitrary dimension [14]. Clifton and Kent extendedthe result further[15] by demonstrating the existence of dense sets of projectionoperators, in complex Hilbert spaces of arbitrary dimension, with the propertythat no two compatible projectors are members of incompatible resolutions ofthe identity. The significance of this property is that it makes it trivial toconstruct a distribution over different hidden states that recovers the quantummechanical expectation values.

CK argue[15] that this construction allows us to define a non-contextualhidden variable theory that simulates quantum mechanics, by the following rea-soning. First, let us suppose that, as in the standard von Neumann formulationof quantum mechanics, every measurement corresponds to a projective decom-position of the identity. However, because any experimental specification of ameasurement has finite precision, we need not suppose that every projectivedecomposition corresponds to a possible measurement. Having defined a denseset of projectors P that gives rise to a dense set of projective decompositions ofthe identity D, we may stipulate that every possible measurement correspondsto a decomposition of the identity in D. The result of any measurement is de-termined by hidden variables that assign a definite value to each operator inP in a non-contextual manner. Via the spectral decomposition theorem, thoseHermitian operators whose eigenvectors correspond to projectors in P are alsoassigned values. If measurements could be specified with infinite precision, thenit would be easy to distinguish this alternative theory from standard quantummechanics. We could simply ensure that our measurements correspond exactlyto the projectors featured in some KS-uncolourable set. If they in fact corre-sponded to slightly different projectors, we would detect the difference.

Now, for any finite precision, and any KS-uncolourable set of projectors,there will be projectors from P sufficiently close that the supposition that our

7For instance, a projective measurement on a quantum system S together with an ancillaryquantum system A requires us, on this view, to take S + A as the system being measured,rather than speaking of a POV measurement being carried out on S.

11

measurements correspond to those from P will not make a detectable difference.So, which particular element of D does this measurement correspond to? CKpropose that the answer to this question is determined algorithmically by thehidden variable theory.

Let us illustrate how this could work. Consider some ordering {d1, d2, . . .}of the countable set D. Let ǫ be a parameter much smaller than the precisionattainable in any current or foreseeable experiment. More precisely, ǫ is suffi-ciently small that it will be impossible to tell from the outcome statistics if ameasurement attempts to measures a decomposition d = {P1, . . . , Pn} and actu-ally measures a decomposition d′ = {P ′

1, . . . , P′

n}, provided |Pi − P ′

i | < ǫ for alli. Suppose now we design a quantum experiment which would, if quantum the-ory were precisely correct, measure the projective decomposition d. (Of course,we can only identify d to within the limits of experimental precision, but, onthe hypothesis that all measurements are fundamentally projective, we supposethat in reality the value of d is an objective fact.) We could imagine that thehidden variable theory uses the following algorithm: first, it identifies the firstdecomposition di = {P i

1, . . . , Pin} in the sequence such that |Pj −P i

j | < ǫ for allj from 1 to n. Then, it reports the outcome of the experiment as that definedby the hidden variables for di: in other words, it reports outcome j if the hiddenvariable theory ascribes value 1 to P i

j (and hence 0 to the other projectors ind).

It may be helpful to visualize this sort of model applied to projectors inthree real dimensions. The system to be measured can be pictured as a spherewith (infinitesimally thin) spines of some fixed length sticking out along all thevectors corresponding to projectors in D, coloured with 1 or 0 at their endpoint.A quantum measurement defines an orthogonal triple of vectors, which in generalis not aligned with an orthogonal triple of spines. Applying the measurementcauses the sphere to rotate slightly, so that a nearby orthogonal triple of spinesbecomes aligned with the measurement vectors. The measurement outcome isthen defined by the spine colourings.

Some points are worth emphasizing here. First, the algorithm we have justdescribed obviously cannot be obtained from standard quantum theory. It isthe hidden variable theory that decides which projective decomposition is ac-tually measured. Some critics have implicitly (or explicitly) assumed that themeasured decomposition must be precisely identified by standard quantum the-oretic calculations.8 But finite precision hidden variable models need not beso constrained: all they need to do is simulate quantum theory to within finiteprecision.

Second, as the algorithm above suggests, any given CK model actually con-tains an infinite collection of sub-models defined by finite subsets {d1, d2, . . . , dr}ofD with the property that they are able to reproduce quantum theory to withinsome finite precision ǫr, where ǫr → 0 as r → ∞. At any given point in time,there is a lower bound on the precision actually attainable in any feasible ex-

8For example Peres[26], whose “challenge” seems to be based on a misunderstanding ofthis point and on neglect of the POV models defined in the next section, and Appleby[28].

12

periment. Hence, at any given point in time, one (in fact infinitely many) ofthe finite sub-models suffices to reproduce quantum theory to within attainableexperimental precision. In other words, at any given point in time, MCK’s ar-gument can be run without using infinite dense subsets of the sets of projectorsand projective decompositions.

Third, we recall that the models CK originally defined are not complete hid-den variable models, since no dynamics was defined for the hidden variables. AsCK noted, the models can be extended to cover sequential measurements sim-ply by assuming that the hidden variables undergo a discontinuous change aftera measurement, so that the probability distribution of the post-measurementhidden variables corresponds to that defined by the post-measurement quan-tum mechanical state vectors. A complete dynamical non-contextual hiddenvariable theory needs to describe successive measurements in which the inter-vening evolution of the quantum state is non-trivial. In fact (though CK didnot note it), this could easily be done, by working in the Heisenberg ratherthan the Schrodinger picture, and applying the CK rules to measurements ofHeisenberg operators. In this version of the CK model, the hidden variables de-fine outcomes for measurements, change discontinuously so as to reproduce theprobability distributions for the transformed quantum state, and then remainconstant until the next measurement.

3.2 Positive operator valued measurements

Dealing with projective measurements is arguably not enough. One quite popu-lar view of quantum theory holds that a correct version of the measurement ruleswould take POV measurements as fundamental, with projective measurementseither as special cases or as idealisations which are never precisely realised inpractice. In order to define a NCHV theory catering for this line of thought,Kent[14] constructed a KS-colourable dense set of positive operators in a com-plex Hilbert space of arbitrary dimension, with the feature that it gives rise toa dense set of POV decompositions of the identity. Clifton and Kent[15] con-structed a dense set of positive operators in complex Hilbert space of arbitrarydimension with the special feature that no positive operator in the set belongsto more than one decomposition of the identity. Again, the resulting set of POVdecompositions is dense, and the special feature ensures that one can averageover hidden states to recover quantum predictions. Each of the three pointsmade at the end of the last section applies equally well to the POV models.

We should stress that the projective and POV hidden variable models definedin Refs. [14, 15] are separate theories. One can consider whichever model oneprefers, depending whether one is most interested in simulating projective orPOV quantum measurements, but they are not meant to be combined. ThePOV hidden variable model does, as of course it must, define outcomes forprojective measurements considered as particular cases of POV measurements— but not in the same way that the projective hidden variable model does.

The CK models for POV measurements have, surprisingly, been neglectedby some critics (e.g. [26]), who object to the CK projective models on the

13

grounds that they unrealistically describe outcomes of ideal but impreciselyspecified projective measurements. As we noted above, this objection is indeedreasonable if one takes the view that one should define the measured quantumsystem in advance, independent of the details of the measurement apparatus,or if one regards POV measurements as fundamental for any other reason. ThePOV measurement models were devised precisely to cover these points.

4 Some criticisms of the MKC models

4.1 Are the CK models classical?

Clifton and Kent claimed that the CK models show “there is no truly compellingargument establishing that non-relativistic quantum mechanics describes clas-sically inexplicable physics”([15], p. 2113). Appleby[28, 30] and others[31] havequeried whether the models can, in fact, properly be described as classical,given that they define values on dense subsets of the set of measurements insuch a way that every neighbourhood contains operators with both truth values— a feature which Appleby characterises[28, 30] as displaying a “radical” or“pathological” discontinuity.9

Our first response is that there is no contradiction here. The models areclassical in the sense of having a phase space structure and the logical structureof a classical theory. Discontinuity in itself is clearly not an obstacle to classi-cality. For one thing, point particles and finite extended objects with boundarydiscontinuities are routinely studied in classical physics. For another, if disconti-nuity of truth values were the crucial issue, the KS theorem would be redundant— it is immediately obvious that any truth values assigned by hidden variablesmust be discontinuous, since the only possible truth values are 0 and 1, andboth must be realised. It is, admittedly, rarer to consider classical systems inwhich discontinuities are dense. But it is not, in our opinion, wrong: givenconsistent evolution laws, one can sensibly study the behaviour of a classicalsystem in which point particles are initially sited at every rational vector in R3,for instance.

To those who find dense discontinuities just too outre, though, we can of-fer an alternative response. As we noted earlier, one can define CK modelswhich simulate quantum mechanics adequately (given any specific attainableexperimental precision) using finite collections of projections and projective de-compositions. These models are still discontinuous, but have only finitely manydiscontinuities. As above, one can visualise such a model, in R3, as defined

9Incidentally, Appleby ([28], p.6) overstates the implications of the CK models’ disconti-nuities. It is true that the CK models give the value of an operator P ′ that is close to butgenerally not identical to the intended operator P , and it is also true that the CK modelsmay ascribe different valuations to P and P ′. However, if an unknown state drawn from aknown ensemble is measured, obtaining the valuation for P ′ generally does give some statis-tical information about the pre-measurement valuation of P . It is thus wrong to imply, asAppleby seems to, that the measurement process in a CK model “does not reveal any moreinformation [. . .] than could be obtained by tossing a coin”.

14

by a sphere with finitely many spines projecting from it. There is no sensibledefinition of classicality that renders such an object (or analogues with moredegrees of freedom) non-classical.

4.2 Are the CK models consistent with quantum proba-

bilities?

In Ref. [32], Cabello argues that any model of the type constructed by CKmust lead to experimental predictions that differ from those of quantum me-chanics. In fact, the CK models are explicitly constructed so as to reproducethe predictions of quantum mechanics, and do so. Cabello’s argument rests onan assumption that is not true of the CK construction. In Ref. [27], Applebyargues that any model of a certain type must either be contextual or violatethe predictions of quantum mechanics. In Ref. [33], Breuer argues that NCHVmodels of yet a different type make different predictions from quantum mechan-ics. Again, Appleby and Breuer both make assumptions that are not true of theCK constructions.

We begin with Cabello’s argument. Cabello makes the following illegitimateassumption about MKC models: if we consider two projectors, P1, P2 ∈ P thatare infinitesimally close to being orthogonal, then for any ensemble of quantumsystems, the proportion of systems for which V (P1) = V (P2) = 1 is infinitesi-mally small. Now of course, in order to reproduce the predictions of quantummechanics, this assumption must hold for ensembles of quantum systems pre-pared in a near eigenstate of P1 or P2. It does not follow, however, that itmust hold for any quantum ensemble. This is correctly noted by Clifton in aprivate communication to Cabello, reported by Cabello in a footnote. Cabello’sresponse is that “from the point of view of an NCHV theory, even if the ensem-ble is prepared in an arbitrary quantum state, any joint measurement...[of P1

and P2]...(which are as close as one desires to being compatible observables) hasa nonzero probability (which increases along with the experiment’s precision)to reveal their predefined values.” In an MKC-type model, however, there isno physical measurement that corresponds to a joint measurement of P1 andP2, as long as they are not precisely orthogonal. Recall the MKC stipulationthat any physical measurement corresponds to a decomposition of the identityfrom the dense set D (where this is a set of projective decompositions or POVdecompositions, depending on the model used). What one can do (in the pro-jective case) is perform a measurement of projectors P ′

1 and P ′

2 that are strictlyorthogonal and are close to P1 and P2 respectively. But it is not a property ofthe CK models that P ′

1 and P ′

2 must have, with high probability, values equalto those of P1 and P2.

Requiring that a hidden variable theory should imply that measurement of anoperator close to P should with high probability reveal a value equal to that of Pmay at first glance seem desirable. Appleby[27, 28, 29, 30] has discussed at somelength this requirement and the fact that MKC models do not incorporate it.We would make one parenthetical comment: the requirement seems to us to beat least partly motivated by a definition of operator distance that is intrinsically

15

quantum mechanical. Two projections P and Q are said to be nearly equal inquantum theory if the operator norm of their difference, |P −Q|, is small. Butthe fact that two projectors are close, by quantum mechanical definitions, doesnot preclude a hidden variable theory from using entirely different quantities todefine the outcomes of the corresponding measurements.

Cabello’s result (along with the results of Ref. [30]) spells out the implicationof the KS theorem that non-contextual hidden variable models which simulatequantum mechanics cannot have the property that nearby projectors alwayshave the same valuation with high probability. However, the evident fact thatthe CK models do not have this feature does not bear on the claim erroneouslycriticised by Cabello: the models are non-contextual and do, as CK claimed,reproduce all the predictions of quantum mechanics.10

Appleby[27] has argued, using assumptions different from Cabello’s, that anynon-contextual model of a certain kind makes different predictions from quan-tum theory. Appleby assumes that in an imprecise measurement of observablescorresponding to three projectors, the three projectors actually measured arenot exactly commuting, but are picked out via independent probability distri-butions. Again, this is not how MKC models work. In an MKC model, if itis the projective type for example, the projectors actually measured are alwayscommuting (this is one of the axioms of the theory that relate its mathematicalstructure to the world, i.e., it is not some kind of miraculous coincidence). Asimilar remark applies to the POV version.

Finally, in Ref. [33], Breuer proves that any finite precision NCHV modelthat assigns values to a dense subset of projection operators, and also satisfiesa certain extra assumption of his own, must make different predictions fromquantum mechanics. Suppose that a spin measurement is performed on a spin-1 particle and that the measurement direction desired by the experimenter (thetarget direction) is ~n. The assumption is that the actual measurement directionis in a random direction ~m, and that the distribution ω~n,ǫ(~m) over possibleactual directions, given ~n and the experimental precision ǫ, satisfies

ωR~n,ǫ(R~m) = ω~n,ǫ(~m),

for all rotations R. Of course, the MKCmodels do not satisfy this condition, andBreuer notes this. In fact, it is clear that no model that colours only a countableset of vectors could satisfy the condition. To those who regard Breuer’s conditionas desirable on aesthetic grounds, we need offer no counter-argument: it wasconceded from the beginning that the CK models are unaesthetic.

4.3 Non-locality and quantum logic

Any hidden variable theory that reproduces the predictions of quantum mechan-ics must be non-local, by Bell’s theorem. The MKC models are no exception.Some have argued [29, 34], however, that non-locality is itself a kind of contextu-ality, and that any theory that is non-local must also, therefore, be contextual.

10Appleby[28] offers essentially the same response to Cabello’s arguments as that given here.

16

Indeed, it is relatively common to read in the literature the claim that non-locality is a special case of contexuality. Here, we simply wish to point out thatnon-locality and contextuality are logically independent concepts. Newtoniangravity provides an example of a theory that is non-contextual and non-local.One can also imagine theories that are contextual and local (for example, a sortof modified quantum mechanics, in which wave function collapse propagates atthe speed of light[35]). Appleby, in Ref. [29], notes the example of Newtoniangravity himself, but states that “in the framework of quantum mechanics thephenomena of contextuality and non-locality are closely connected.” This istrue, but it is not necessarily the case that what is true in the framework ofquantum mechanics is still true when we take the point of view of the hiddenvariables — and when assessing hidden variable models, it is the hidden vari-ables’ point of view that is important. Appleby concludes, based on a GHZ-typeexample, that the MKC models display “existential contexuality”. It seems tous that, considered from the proper hidden variable model theoretic rather thanquantum theoretic perspective, Appleby’s argument simply demonstrates thenon-locality of the CK models — which were, of course, explicitly presented[15]as non-relativistic and necessarily non-local.

Finally, some have objected to the MKCmodels on the grounds that elementsof the quantum formalism, for example the superposition principle [17] or thequantum logical relations between projectors [31, 36], are not preserved. Wenote that this is of no importance from the point of view of the hidden variables.The whole point is that they have their own classical logical structure.

5 The operational approach

In Sec. 2, we discussed some proposed experiments, intended as tests of theKochen-Specker theorem. We pointed out that there is an important disanal-ogy with Bell experiments, in that in the Kochen-Specker case, one always hasthe logical possibility of rejecting Hilbert space structure and constructing a newtheory that is non-contextual in its own terms. Hence one could never concludeunequivocally that Nature is contextual. On the other hand, if one accepts thestandard quantum formalism, and is seeking to “complete” the description withhidden variables, then no experiment is needed, as a non-contextual interpre-tation can be ruled out on logical grounds alone. An experiment could onlyserve to rule out a non-contextual theory that accepts at least part of Hilbertspace structure, but rejects, for example, the KS criteria. Even then, the MKCmodels would exhibit a loophole in such an experiment.

Having said this, we noted at the end of Sec. 2 that one can try to framea definition of contextuality that is independent of Hilbert space structure, bygiving a completely operational definition of “observable” and hence of “contex-tuality”. Can we not then perform an experiment that will allow us to concludethat Nature is contextual? And if so, what of the MKCmodels? The operationalapproach is hinted at in Ref. [16] and worked out explicitly by Simon, Brucknerand Zeilinger (SBZ) [24] and Larsson [25]. The work of both SBZ and Larsson

17

is motivated by the issue of finite precision and is presented as a riposte toMKC. SBZ, for example, describe their work as showing “how to derive hidden-variable theorems that apply to real experiments, so that non-contextual hiddenvariables can indeed be experimentally disproved.” This seems to contradict di-rectly the claims of MKC, who say that the MKC models are non-contextualand reproduce correctly the quantum predictions for any finite precision experi-ment. We shall see, however, that there is really no tension here. The apparentcontradiction rests on different uses of the word “contextual”. Further, we shallargue that the work of SBZ and Larsson, while interesting, does not have thesignificance they claim. For definiteness, we discuss the work of SBZ, althoughLarsson’s is very similar.

SBZ consider a black box with three knobs, each of which has a finite num-ber of different settings. After setting the knobs, an observer presses a “go”button. He then receives an outcome for each knob, which is either a 1 or a0. As an example of such a box, we can consider one that contains within it aquantum experiment in which the spin squared of a spin-1 particle is measuredin three different directions. The directions are determined to some degree ofaccuracy by the settings of the knobs. However, it will not be the case that agiven knob setting corresponds to a measurement of spin squared in preciselythe same direction every time the box is used. There will be experimental in-accuracies. In general, we may imagine that there are some hidden variablesassociated with the measuring apparatus, as well as the quantum system, whichdetermine exactly what measurement is being performed. From the point ofview of our observer outside the black box, however, none of this matters. Allhe has access to are the three knobs and the outcomes. SBZ propose that theobserver should simply, by fiat, define observables operationally, with each ob-servable corresponding to a different setting of one of the knobs. He can alwaysbe sure which observable he is measuring, according to this operational defini-tion, even though he cannot be sure which observable is actually being measuredaccording to quantum theory.

Not knowing what is happening inside the box, our outside observer cantry to formulate a model theory. In a deterministic model theory, the entireinside of the box can be described by some hidden state that predicts what thethree outcomes will be for each possible joint setting of the knobs. The modelis non-contextual if, for each hidden state, the outcome obtained for each knobdepends only on its setting, and not on the settings of the other two knobs. Onrunning the box repeatedly, the observer can build up outcome statistics foreach possible joint knob setting. If no non-contextual model of the workings ofthe box that reproduces these statistics exists, then, SBZ propose, we shouldsay that the box is “contextual”.

For instance, consider a set of 3-dimensional vectors that is KS-uncolourable,in the sense that it is impossible to give each vector a 0 or a 1 such that eachorthogonal triad consists of one 1 and two 0s. The set of vectors can be written,for example

{{~n1, ~n2, ~n3}, {~n1, ~n4, ~n5}, . . .}.

18

For the set to be KS-uncolourable, it must be the case that some vectors appearin more than one triad. Suppose that these triads are taken to indicate possibletriads of knob settings. Suppose that the experiment is run many times, and itis found that whenever one of these triads is measured, the outcomes consist ofone 1 and two 0s. Then we can conclude, from the fact that the set of vectorsis KS-uncolourable, that the box is “contextual” according to SBZ’s definition— a property we refer to hereafter as SBZ-contextual.

This, though, is too much of an idealisation. In a real experiment therewill be noise, which will sometimes cause non-standard results, for exampletwo 1s and a 0. The core of SBZ’s paper is a proof of the following result.Imagine that the box is run many times, with knob settings corresponding toorthogonal triads, and that the outcomes are one 1 and two 0s in a fraction1 − ǫ of cases. Then, the box must be SBZ-contextual if ǫ < 1/N , where N isthe number of orthogonal triads appearing in the set. If the box is in fact aquantum experiment in which the spin squared of a spin-1 particle is measuredin different directions, then increasing the accuracy of the experiment will beable to reduce ǫ below 1/N . The observer will be able to conclude that theexperiment is SBZ-contextual.

We wish to make several related remarks concerning this result. The firstthing is to clarify the implication for MKC models. A box with a quantum spinexperiment inside is certainly simulable by an MKC model, since the modelsare explicitly constructed to reproduce all the predictions of quantum mechanicsfor finite precision measurements. How will the simulation work? On each run,the knob settings determine approximately which measurement is performed,but exactly which is determined randomly, or by apparatus hidden variables.The exact measurement corresponds to some Hermitian operator in the MKCKS-colourable set. The outcome is determined by a hidden state that assignsa definite value to each operator in the KS-colourable set in a non-contextualmanner. Hence if observables are defined by operators, it is true that the valueobtained on measuring a given observable does not depend on which other ob-servables are measured at the same time and in this sense, the MKC modelis non-contextual. The fact that the black box is SBZ-contextual tells us thatthe settings of all three knobs together, along with the apparatus hidden state,are needed to determine the Hermitian operators that are in fact being mea-sured. In a way, of course, it couldn’t be any different, since one cannot expectan algorithm that chooses three vectors independently to generally produce anorthogonal triad. The SBZ-contextuality of the black box tells us in additionthat for at least some apparatus hidden states, whether the measurement cor-responds to a triad for which knob i gets outcome 0 or a triad for which knob igets outcome 1 depends on the settings of knobs j and k.

This should be enough to show that there is no formal contradiction betweenthe MKC and the SBZ results. Some may argue, however, that from a physicalpoint of view, the operational definition of SBZ-contextuality is the only inter-esting one, and that the MKC models, therefore, are not non-contextual in anyinteresting sense — or at least that the operational definition is an interestingone, and the MKC models are not non-contextual in this sense. We wish to

19

counter such arguments with some cautionary remarks concerning these blackboxes.

First, SBZ, as did the authors of the experiments discussed in Sec. 2 above,motivate their work via an analogy with Bell’s theorem. The disanalogy we men-tioned in Sec. 2 has disappeared now that observables are defined operationally.However, there is another important disanalogy. This is that there is nothingspecifically non-classical about a black box that is behaving SBZ-contextually.One could easily construct such a box out of cog-wheels and springs. Thus withno knowledge of or assumptions about the internal workings of the box, onecould not use it to distinguish classical from quantum behaviour. This shouldbe contrasted rigorously with the case of a non-local black box. If a (long thin)black box is seen to be behaving non-locally, then we know that we are in aquantum, and not a classical, universe. Such a box can even be used for infor-mation theoretic tasks that cannot be accomplished classically (e.g. [37]). Givena black box that is SBZ-contextual, we have no such guarantees. This seems tous to cast doubt on the use or significance of a purely operational definition ofcontextuality, as opposed to a theory-relative one.

Second, the fact of the matter is that any realistic experiment, whethercarried out in a classical or a quantum universe, will necessarily exhibit SBZ-contextuality to some (possibly tiny) degree. Not only that, the outcome proba-

bilities for any given SBZ-observable will depend (at least slightly) on the contextof the other knob settings. On moving one knob, for example, its gravitationalfield will be changed, and this will affect the behaviour of the whole apparatus.This is not a consequence of quantum theory. It would be true of an experimentin which a classical measuring apparatus measures classical observables on aclassical system. Yet we would not infer from this SBZ-contextuality of the out-comes that classical physics is (at least slightly) contextual. We do not take SBZand Larsson to be advocating otherwise: all sides in the Kochen-Specker debateagree that classical physics is, paradigmatically, non-contextual. Rather, wetake the fact that the opposite conclusion follows from SBZ’s and Larsson’s def-initions to indicate that the definition of SBZ-contextuality is inherently flawed.Similarly, we take the fact that SBZ’s definition of an observable can in principleempirically be shown to be context-dependent — since the outcome probabil-ities depend at least slightly on knob settings that are meant to correspondto independent observables — to be a fatal flaw in that definition. An SBZ-observable turns out, under scrutiny, to be a rather complicated construct, withquite different properties from its quantum namesake. A less freighted name— “dial setting”, for instance — would make clearer the obstacles which SBZwould need to surmount in order even to begin to a properly founded discussionof finite precision experimental tests of contextuality.

This last point really needs no reinforcement, but it can be reinforced. Con-sider again the black box that in fact contains a quantum experiment in whichthe spin squared of a spin-1 particle is measured in different directions. Theidea was to run the box repeatedly with certain combinations of knob settingsthat correspond to the orthogonal triads in a KS-uncolourable set of vectors.However, assuming that they can be moved independently, there is nothing to

20

stop us from setting the knobs in any combination of settings, in particular,in combinations that correspond to triads of non-orthogonal vectors from theKS-uncolourable set. What would happen in this case? The quantum exper-iment inside the box cannot be effecting a simultaneous measurement of thespin squared in three directions approximating the knob settings, because thesespin squared observables will not be co-measurable. Perhaps the box measuresspin squared in three orthogonal directions, at least one of which is not close tothe corresponding knob setting. Or perhaps the box does some kind of positiveoperator-valued measurement. In either case, it seems that for most quantumexperiments, from the observer’s point of view, the outcomes will inevitably becontextual even at the level of the quantum probabilities, and even if we un-realistically neglect the classical perturbations produced on the apparatus byaltering any of the knobs. Given that the box is behaving in an overtly contex-tual manner even at the level of probabilities, one is then again led to ask: whyshould we be interested in whether the box can be described in a non-contextualfashion in the special case that we carefully restrict our knob settings so thatthey always correspond to orthogonal triads in the KS-uncolourable set?

Taking these points on board, careful operationalists might try to refine theirposition by speaking, not of a distinction between SBZ-contextuality and SBZ-non-contextuality, but instead of degrees of SBZ-contextuality. It could be ar-gued that, although classical mechanics is indeed SBZ-contextual in SBZ’s sense,the perturbations which imply SBZ-contextualities in outcome probabilities willgenerally be very small, and the outcome probability SBZ-contextualities cor-respondingly hard to detect: indeed, in principle, with sufficient care, the per-turbations can be made as small as desired. In contrast, SBZ and Larsson’sresults might be interpreted as implying that quantum experiments display anirreducible finite degree of SBZ-contextuality a la SBZ. The difficulty with thisline of argument is that, as the CK models illustrate, it is not always true in clas-sical mechanics that small perturbations induce (only) correspondingly subtleeffects. Operationalists need to frame a definition separating classical mechan-ics from the CK models in order to maintain that the former theory is at leastapproximately or effectively SBZ-non-contextual and the latter is definitivelySBZ-contextual. This cannot be done: as we have already noted, the MKCmodels show in principle how to build classical devices which non-contextuallysimulate quantum theory up to any given fixed nonzero precision.

In summary, even black box operational definitions do not allow unambigu-ous experimental discrimination between contextual and non-contextual the-ories, and thus present no challenge to MKC’s assertion that non-contextualtheories can account for current physics. SBZ’s operational definition of SBZ-contextuality does give us a clear, theory independent notion of something, butit is not contextuality in any sense consistent with standard usage. In particu-lar, the notion defined is not able to separate the properties of quantum theoryand classical mechanics, and so is not of fundamental relevance to the debateover finite precision and the KS theorem. Attractive though it would be to de-vise a sensible theory-independent definition of (non-)contextuality, we do notbelieve it is possible. We see no fundamentally satisfactory alternative to re-

21

stricting ourselves to talking of theories as being non-contextual or contextual,and using theory-relative definitions of these terms.

6 A Closing Comment

We think it appropriate to stress here that neither the preceding discussion norRefs. [14, 15] are or were intended to cast doubt on the essential importance andinterest of the Kochen-Specker theorem. As we have stressed throughout, ourinterest in examining the logical possibility of non-contextual hidden variablessimulating quantum mechanics is simply that it is a logical — albeit scientificallyhighly implausible — possibility, which demonstrates interesting limitations onwhat we can rigorously infer about fundamental physics.

Acknowledgements

We are profoundly indebted to Rob Clifton for many lively and stimulat-ing discussions and much encouragement. We also warmly thank Marcus Ap-pleby, Jeremy Butterfield, Adan Cabello, Chris Fuchs, Sheldon Goldstein, Lu-cien Hardy, Jan-Ake Larsson, David Mermin, David Meyer, Asher Peres andChristoph Simon for very helpful discussions and criticisms.

References

[1] E. Specker, Die Logik Nicht Gleichzeitig Entsheidbarer Aussagen, Dialectica14 239-246 (1960).

[2] S. Kochen and E. Specker, The Problem of Hidden Variables in Quantum

Mechanics, J. Math. Mech. 17 59-87 (1967).

[3] J. S. Bell, On the Problem of Hidden Variables in Quantum Mechanics, Rev.Mod. Phys. 38 447-452 (1966), reprinted in Speakable and Unspeakable in

Quantum Mechanics (Cambridge University Press, 1987).

[4] A. Peres, Quantum Theory: Concepts and Methods (Kluwer Academic Pub-lishers; Boston MA; 1995); Chap. 7, pp. 187-211.

[5] J. Zimba and R. Penrose, On Bell Non-locality without probabilities: more

curious geometry, Stud. Hist. Phil. Sci. 24 697-720 (1993).

[6] J. Conway and S. Kochen, unpublished.

[7] A. Cabello, J. Estebaranz and G. Garcia Alcaine, Bell-Kochen-Specker the-

orem: A proof with 18 vectors, Phys. Lett. A 212 183 (1996).

[8] I. Pitowsky, Resolution of the Einstein-Podolsky-Rosen and Bell Paradoxes,Phys. Rev. Lett. 48 1299-1302 (1982); Pitowsky Responds, Phys. Rev. Lett.49 1216 (1982).

22

[9] I. Pitowsky, Deterministic Model of Spin and Statistics, Phys. Rev. D 27

2316-2326 (1983).

[10] I. Pitowsky, Discussion: Quantum Mechanics and Value Definiteness, Phil.Sci. 52 154-156 (1985).

[11] A. Gleason, Measures on Closed Subspaces of Hilbert Space, J. Math. Mech.6 885-893 (1957).

[12] D. Meyer, Finite Precision Measurement Nullifies the Kochen-Specker The-

orem, Phys.Rev.Lett. 83 3751-3754 (1999).

[13] C. Godsil and J. Zaks, Coloring the sphere, University of Waterloo researchreport CORR 88-12 (1988).

[14] A. Kent, Non-Contextual Hidden Variables and Physical Measurements,Phys. Rev. Lett. 83 3755-3757 (1999).

[15] R. Clifton and A. Kent, Simulating Quantum Mechanics by Non-Contextual

Hidden Variables, Proc. Roy. Soc. Lond. A 456, 2101-2114 (2000).

[16] N. D. Mermin, A Kochen-Specker Theorem for Imprecisely Specified Mea-

surements, quant-ph/9912081.

[17] A. Cabello, Comment on “Non-Contextual Hidden Variables and Physical

Measurements” , quant-ph/9911024.

[18] A. Cabello and G. Garcia-Alcaine, Proposed experimental tests of the Bell-

Kochen-Specker theorem, Phys. Rev. Lett. 80 1797 (1998).

[19] C. Simon et al., A feasible “Kochen-Specker” experiment with single parti-

cles, Phys. Rev. Lett. 85, 1783 (2000).

[20] J.S. Bell, Free variables and Local Causality, Epistemological Letters(February 1977); reprinted in Dialectica 39 103 (1985).

[21] S. Basu et al., A New Quantum Violation of Noncontextuality in-

voking Bell’s inequality and Entangled State of a Spin- 12

Particle,quant-ph/9907030.

[22] N.D. Mermin, quoted in Ref. [18].

[23] B. van Fraassen, Semantic Analysis of Quantum Logic, in Contemporary

Research in the Foundations and Philosophy of Quantum Theory (Dor-drecht, Reidel, 1973); see also Chapter 5 of M. Redhead, Incompleteness,

Nonlocality and Realism, Oxford University Press (Oxford, 1987).

[24] C. Simon, C. Brukner and A. Zeilinger, Hidden Variable Theorems for Real

Experiments, Phys. Rev. Lett. 86 4427-4430 (2001).

[25] J-A. Larsson, A Kochen-Specker Inequality, Europhys. Lett., 58 (6) 799-805(2002).

23

[26] A. Peres, What’s Wrong with these Observables?, quant-ph/0207020.

[27] D. Appleby, Contextuality of Approximate Measurements,quant-ph/0005010.

[28] D. Appleby, Nullification of the Nullification, quant-ph/0109034.

[29] D. Appleby, Existential Contextuality and the Models of Meyer, Kent and

Clifton, Phys. Rev. A 65, 022105 (2002).

[30] D. Appleby, The Bell-Kochen-Specker Theorem, quant-ph/0308114.

[31] H. Havlicek et al., On colouring the rational quantum sphere, J. Phys. A:Math. Gen. 34 (14), 3071-3077 (2001).

[32] A. Cabello, Finite-precision measurement does not nullify the Kochen-

Specker theorem, Phys. Rev. A 65 052101 (2002).

[33] T. Breuer, Kochen-Specker Theorem for Finite Precision Spin One Mea-

surements, Phys. Rev. Lett. 88 240402 (2002).

[34] C. F. Boyle and R. L. Schafir, Remarks on “Noncontextual hidden variables

and Physical Measurements”, quant-ph/0106040.

[35] A. Kent, Causal Quantum Theory and the Collapse Locality Loophole,quant-ph/0204104.

[36] P. Busch, Quantum states and generalized observables: a simple proof of

Gleason’s theorem, quant-ph/9909073.

[37] H. Buhrman, R. Cleve and W. van Dam, Quantum Entanglement and Com-

munication Complexity, quant-ph/9705033.

24