The status of determinism in proofs of the impossibility of a noncontextual model of quantum theory
aa r X i v : . [ qu a n t - ph ] J a n The status of determinism in proofs of the impossibility of a noncontextual model ofquantum theory
Robert W. Spekkens ∗ Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario Canada N2L 2Y5 (Dated: Jan. 5, 2015)In order to claim that one has experimentally tested whether a noncontextual ontological modelcould underlie certain measurement statistics in quantum theory, it is necessary to have a notion ofnoncontextuality that applies to unsharp measurements, i.e., those that can only be represented bypositive operator-valued measures rather than projection-valued measures. This is because any re-alistic measurement necessarily has some nonvanishing amount of noise and therefore never achievesthe ideal of sharpness. Assuming a generalized notion of noncontextuality that applies to arbitraryexperimental procedures, it is shown that the outcome of a measurement depends deterministi-cally on the ontic state of the system being measured if and only if the measurement is sharp.Hence for every unsharp measurement, its outcome necessarily has an in deterministic dependenceon the ontic state. We defend this proposal against alternatives. In particular, we demonstrate whyconsiderations parallel to Fine’s theorem do not challenge this conclusion. I. INTRODUCTION
The Bell-Kochen-Specker theorem concerns the pos-sibility of explaining the predictions of quantum theoryin terms of a noncontextual ontological model [1, 2]. Inrecent years, there has been some effort devoted to mak-ing the notion of a noncontextual ontological model moreoperational [4–7]. The goal of such efforts is to be ableto decide, for any given operational theory (possibly dis-tinct from quantum theory), whether it admits of such amodel or not. Achieving this goal would allow one to as-sess the possibility of a noncontextual ontological modelbased solely on experimental data (regardless of whetheror not this data is consistent with quantum theory): justas the observation of Bell inequality violations [8] ruleout a locally causal model of nature , irrespective of thetruth of quantum theory, so too would violations of oper-ational noncontextuality inequalities rule out a noncon-textual ontological model of nature.This article does not explicitly address the question ofhow to derive such inequalities. Rather, it presents somepreparatory work towards this goal. We presume thecorrectness of operational quantum theory, but we con-sider the predictions of quantum theory for experimentsthat do not satisfy the idealizations that are typically as-sumed in discussions of noncontextuality. In particular,we relax the assumption that all the measurements in theexperiment are projective. This assumption is an ideal-ization because any realistic measurement procedure ona physical system is subject to some unavoidable noise,and consequently cannot be represented by a set of pro-jectors on the Hilbert space of the system. Rather, itmust be represented by a positive operator valued mea- ∗ Electronic address: [email protected] More precisely, they rule out models that satisfy Bell’s notion oflocal causality and certain other assumptions, such as the freedomof settings. sure (POVM), that is, a set of positive operators (called effects ) that sum to the identity operator.Recall that any measurement procedure can be repre-sented by a POVM. In the special case where all of theeffects in the POVM are projectors, we call the POVMa projector-valued measure and we call the measurementprojective or sharp . If the POVM that represents a mea-surement is not a projector-valued measure, we call themeasurement unsharp . The fact that no realistic mea-surement procedure is strictly free of noise implies thatsharpness is an ideal that is never actually realized inany experiment. Consequently, the question of how torepresent unsharp measurements in a noncontextual on-tological model is critical if one wishes to compare thepredictions of such a model to the predictions of quan-tum theory that actually arise in realistic experiments.One proposal which has been made is that every un-sharp measurement should be represented by a set of outcome-deterministic response functions. What thismeans is that the ontological model specifies hidden vari-ables which fix the outcome of each unsharp measure-ment (in a context-independent manner). This assump-tion, of outcome determinism for all unsharp measure-ments , will be abbreviated as ODUM. It is explicit, forinstance, in Ref. [9] (here v ( E ) denotes the value assignedby the hidden variables to an effect E appearing in aPOVM):An interpretation of valuations as truth valueassignments would require the numbers v ( E )to be either 1 or 0, indicating the occurrenceor nonoccurrence of an outcome associatedwith E . Valuations with this property arereferred to as dispersion-free.The assumption of ODUM is also made in Ref. [10] (here,the discussion is of a particular set of unsharp quantummeasurements):Each [POVM in the set] contains eightpositive-semidefinite operators whose sum isthe identity. Therefore, a noncontextualhidden-variable theory must assign the an-swer yes to one and only one of these eightoperators.In fact, this assumption has been made in almost everyarticle that has considered noncontextuality for unsharpmeasurements [9–15].In this article, we will argue against ODUM.Adopting the operational notion of a noncontextualontological model that the author proposed in Ref. [7],we will argue that unsharp measurements in quantumtheory must be represented by outcome- in deterministicresponse functions. The representation is still noncon-textual insofar as the probability that one assigns to anoutcome does not depend on the context. The repre-sentation also remains highly constrained because sharpmeasurements will still need to be represented by a setof outcome-deterministic response functions (See Thm. 3below). This fact about the representation of sharpmeasurements is relevant for realistic experiments, eventhough all the measurements in such experiments areunsharp. This is because quantum theory dictates thata given unsharp measurement can be related in variousways to a set of idealized sharp measurements. For in-stance, it may be the case that the unsharp measurementrealized in an experiment is a convex mixture of a set ofidealized sharp measurements. As another example, itis always possible to express an unsharp measurementon some system as the reduction of an idealized sharpmeasurement on a larger system. Such relations typi-cally imply that the response function for a given unsharpmeasurement is expressible in terms of the response func-tions for the idealized sharp measurements; the fact thatthe latter are required to be outcome-deterministic thenimposes nontrivial constraints on the former.We begin by highlighting an intuitive (but mistaken)argument in favour of ODUM. It is well-known in foun-dational circles that if one can find a local indetermin-istic ontological model for some set of correlations in aBell-type experiment, then one can also find a local de-terministic ontological model for the same set. This wasfirst noted by Fine [16]. As such, if one can rule out local The fact that all noncontextual ontological models must repre-sent sharp measurements outcome-deterministically implies in par-ticular that the ψ -complete ontological model of quantum theoryfails to be a noncontextual model. The ψ -complete ontologicalmodel of quantum theory is the one wherein the ontic states arethe pure quantum states (i.e. no hidden variables) and the re-sponse function associated to an effect is simply the conditionalprobability defined by the Born rule [19]. This model is sometimescalled the “orthodox” interpretation of quantum theory. Giventhe use of the Born rule, the response functions in this model areoutcome-indeterministic even for sharp measurements and there-fore our result implies that the model cannot be noncontextual.This inference is correct because the ψ -complete ontological modelfails to be preparation noncontextual , as explained in Sec. VIII.Bof Ref. [7]. deterministic models for some set of correlations, thenone has also ruled out local indeterministic models. Inother words, although one might have imagined that theclass of correlations that can be explained by local inde-terministic models is strictly larger than the class thatcan be explained by local deterministic models, it turnsout that the classes of correlations that can be explainedby the two sorts of models are precisely the same.In discussions of noncontextuality, the role of deter-minism is less clear. Kochen and Specker’s notion of anoncontextual model explicitly incorporated the assump-tion that the outcomes of a measurement should be fixeddeterministically by the hidden variables. One of theselling points of the operational definition of noncontex-tuality proposed in Ref. [7] is that it explicitly disentan-gles the notion of measurement noncontextuality fromthe assumption of outcome determinism. Furthermore,by using a notion of noncontextuality for preparations ,one can justify the assumption of outcome determinismfor sharp measurements [7]. As such, the contradictionthat is derived in the Kochen-Specker theorem forces usto reject some assumption of noncontextuality. However,there is nothing that warrants assuming outcome deter-minism for unsharp measurements. As such, any no-gotheorem that assumes ODUM and derives a contradictiondoes not deserve to be called a proof of the impossibilityof a noncontextual ontological model because in the faceof a contradiction, one can always assume that the faultyassumption was that of outcome determinism rather thanthat of noncontextuality.Nonetheless, one might wonder whether a version ofFine’s theorem applies to noncontextual models just asit does to locally-causal models. That is, one might won-der whether arguments analogous to those of Fine couldestablish that one can rule out noncontextual ontologi-cal models that are outcome-indeterministic for unsharpmeasurements if and only if one can also rule out thosethat are outcome-deterministic for these measurements.If this were so, then there would be no loss of generalityin assuming ODUM, and consequently no good reasonnot to do so.In fact, in Ref. [17], this is the reason given for notconsidering outcome-indeterministic models:[We] do not discuss stochastic hidden variabletheories explicitly. This does not limit thegenerality of the results derived because theexistence of a stochastic local or noncontex-tual hidden variable model for a given physi-cal system implies that also an underlying de-terministic local or noncontextual model canbe constructed which reproduces the proba-bilities of the stochastic model. Therefore,e.g., ruling out all possible noncontextual de-terministic hidden variable models impliesruling out all possible noncontextual stochas-tic models as well.In this article, we will explain why this intuition ismistaken, that is, why Fine’s theorem does not extend tononcontextual models in the manner that would be re-quired to justify ODUM. The explanation, in essence, isthe following: while it is true that one can always modela measurement that depends indeterministically on theontic state of the system by one that depends determin-istically on the ontic state of a larger system (including,for instance, hidden variables in the apparatus), never-theless, under such a change of representation one findsthat measurement events that differ only by a choice ofcontext are mapped to measurement events that differin their operational statistics and consequently one losesthe warrant to apply the assumption of noncontextuality.We also provide a number of additional argumentsagainst outcome-deterministic representations of unsharpmeasurements in noncontextual models and against vari-ants of these arguments that seek to limit the sorts ofunsharp measurements to which the assumption of non-contextuality can be applied.In addition, we discuss some of the consequences ofour result that measurements are represented outcome-deterministically if and only if they are sharp. For in-stance, the assumption of measurement noncontextual-ity implies the existence of a valuation that assigns toeach effect a probability (independently of the POVM inwhich that effect appears), and our results imply that thevalue assigned to an effect must be one of its eigenval-ues. This observation allows us to provide significantlysimplified versions of two proofs of the impossibility of anoncontextual ontological model of operational quantumtheory which were first presented in Ref. [7] and whichmake explicit use of the assumption of noncontextualityfor unsharp measurements.The article is organized as follows. In Sec. II, weprovide definitions of the relevant background conceptsfor this article: operational theories, ontological models,noncontextuality, and outcome determinism. Sec. III de-scribes the intuition in favour of ODUM (along the linesof Fine’s theorem) and why it is mistaken. Additionalsupporting material for this section is provided in the ap-pendices. Sec. IV presents the proof that within a non-contextual ontological model, a measurement is repre-sented outcome-deterministically if and only if it is sharp.We also discuss which predictions of operational quantumtheory are required for the proof and the significance ofthe result. In Sec. V, we discuss how post-processingof a measurement is represented in an ontological modeland draw out the consequences for noncontextual repre-sentations of sharp and unsharp measurements. In par-ticular, we formulate the constraints on noncontextualrepresentations of measurements in terms of probability-assignments to effects. Sec. VI presents a few more argu-ments against ODUM, in the form of a dialogue betweenthe author and an imaginary proponent of ODUM. Fi-nally, in Sec. VII, we use the constraints on noncontextualrepresentations of measurements in terms of probabilityassignments to effects (introduced in Sec. V) to providesimple proofs of no-go theorems for noncontextuality that make explicit use of unsharp measurements. II. GENERALIZED NONCONTEXTUALITYAND OUTCOME-DETERMINISM An operational theory is a triple ( P , M , p ) where P isa set of preparation procedures, an element of which isdenoted by P , where M is a set of measurement pro-cedures, an element of which is denoted by M and forwhich the corresponding set of outcomes (assumed hereto be discrete) is denoted by K M , and where p : P ×M → [0 , K M :: ( P, M ) p ( k | M, P ) is a nonnegative functionsatisfying P k ∈K M p ( k | M, P ) = 1 for all M ∈ M and all P ∈ P . p ( k | M, P ) specifies the probability of obtainingoutcome k in a measurement of M following a prepara-tion P. In brief, an operational theory specifies the pos-sible measurements and preparations and the statisticsone obtains over the outcomes of any measurement givenany preparation.An ontological model for an operational theory( P , M , p ) is a triple (Λ , µ, ξ ) where Λ is a discrete set ,where µ : P × Λ → [0 ,
1] :: (
P, λ ) µ ( λ | P ) is a non-negative function satisfying P λ ∈ Λ µ ( λ | P ) = 1 for all P ∈ P , where ξ : M × Λ → [0 , K M :: ( M, λ ) ξ ( k | λ, M ) is a nonnegative vector function satisfying P k ∈K M ξ ( k | λ, M ) = 1 for all λ ∈ Λ and for all M ∈ M , and where X λ ∈ Λ µ ( λ | P ) ξ ( k | λ, M ) = p ( k | M, P ) , (1)for all M ∈ M and P ∈ P . The interpretation of thesemathematical expressions is as follows: λ ∈ Λ denotesan ontic state , that is, a complete specification of all thephysical properties of the system (including a valuationof any hidden variables), and Λ is the space of possibleontic states for the system; µ ( λ | P ) is the probability thatthe system is in ontic state λ given that it was preparedaccording to preparation P , and ξ ( k | λ, M ) is the proba-bility that measurement M yields outcome k given thatthe system which was fed into the measurement devicehad ontic state λ. The set { ξ ( k | λ, M ) : k ∈ K M } , consid-ered as functions over Λ, will be called the set of responsefunctions associated with M . Equation (1) asserts thatthe overall probability of outcome k given preparation P and measurement M specified by the operational theoryis recovered by the ontological model.According to the proposal of Ref. [7], noncontextualityis a property of an ontological model of an operationaltheory. A distinction is also made between two sorts ofnoncontextuality, one which pertains to the representa- More generally, an ontological model may be defined such thatΛ is a continuous space, in which case µ : P × Λ → R + is a proba-bility density and sums over λ are replaced by integrals. tion of preparations and the other of which pertains tothe representation of measurements. Definition 1 (noncontextuality)
An ontologicalmodel (Λ , µ, ξ ) of an operational theory ( P , M , p ) issaid to be preparation noncontextual if the followingimplication holds: for any pair of preparation procedures P, P ′ ∈ P ,if p ( k | P, M ) = p ( k | P ′ , M ) ∀ M ∈ M then µ ( λ | P ) = µ ( λ | P ′ ) ∀ λ ∈ Λ . (2) In other words, preparation noncontextuality is the as-sumption that if two preparations yield the same statisticsfor all possible measurements then they are representedequivalently in the ontological model.Similarly, an ontological model of an operational the-ory is said to be measurement noncontextual if the fol-lowing implication holds: for any pair of measurementprocedures
M, M ′ ∈ M ,if p ( k | P, M ) = p ( k | P, M ′ ) ∀ P ∈ P then ξ ( k | λ, M ) = ξ ( k | λ, M ′ ) ∀ λ ∈ Λ . (3) In other words, measurement noncontextuality is the as-sumption that if two measurements have the same statis-tics for all possible preparations then they are representedequivalently in the ontological model.
It is the notion of measurement noncontextuality thatwill be of particular interest to us here.In order to understand this notion better, it is use-ful to adopt the following terminological conventions.In an operational theory, the operational statistics of ameasurement M are the probability distributions overoutcomes for all possible preparation procedures , thatis, the set { ( p ( k | M, P ) : k ∈ K M ) : P ∈ P} . If two mea-surements M and M ′ have the same operational statis-tics, then they are said to be statistically indistinguish-able relative to all preparations . In an ontological modelof an operational theory, the ontological statistics of ameasurement M are the probability distributions overoutcomes for all possible ontic states λ, that is, the set { ( ξ ( k | M, λ ) : k ∈ K M ) : λ ∈ Λ } . If two measurements M and M ′ have the same ontological statistics, then theyare said to be statistically indistinguishable relative toall ontic states . Given this terminology, the assumptionof measurement noncontextuality can be summarized assimply:An ontological model of an operational the-ory is measurement noncontextual if mea-surements that are statistically indistinguish-able relative to all preparations are statis-tically indistinguishable relative to all onticstates.We now provide a definition of the notion of outcomedeterminism, the central topic of this article. We heremake use of the notion of a measurement event , whichrefers to the pair consisting of a measurement and anoutcome of that measurement. Definition 2 (outcome determinism)
A responsefunction ξ ( k | λ, M ) representing a measurement event(outcome k of measurement M ) is said to be outcome-deterministic if ξ ( k | λ, M ) ∈ { , } ∀ λ ∈ Λ , (4) that is, if the probability of obtaining the outcome in ques-tion, conditioned on the ontic state λ , is either or .Otherwise, it is said to be outcome-indeterministic. Ameasurement M associated to a set of response functions { ξ ( k | λ, M ) : k ∈ K M } is said to be represented outcome-deterministically if every response function in the set isoutcome-deterministic. Otherwise, it is said to be repre-sented outcome-indeterministically. We will also say that we have outcome determinism(respectively outcome indeterminism) for some class ofmeasurements if every measurement in the class is repre-sented outcome-deterministically (respectively outcome-indeterministically). We can therefore enumerate the three possibilities forhow ontological models of quantum theory might treatsharp measurements:(a) Outcome determinism for all sharp measurements(b) Outcome indeterminism for all sharp measurements(c) Outcome determinism for some sharp measure-ments and outcome indeterminism for othersand the three possibilities for how they might treat un-sharp measurements(a ′ ) Outcome determinism for all unsharp measure-ments (abbreviated ODUM)(b ′ ) Outcome indeterminism for all unsharp measure-ments(c ′ ) Outcome determinism for some unsharp measure-ments and outcome indeterminism for othersIt turns out that for ontological models of operationalquantum theory, a noncontextual representation of sharpmeasurements must be outcome-deterministic, i.e., op-tion (a), and a noncontextual representation of unsharpmeasurements must be outcome-indeterministic, i.e., op-tion (b ′ ). Indeed, this is the main positive result of ourarticle, which we summarize in the following theorem. There is a subtlety in our choice of terminology which mightlead to confusion. For a set of response functions to be consid-ered outcome-indeterministic, it is sufficient for a single responsefunction in the set to be outcome-indeterministic. Hence, every setof response functions is either outcome-deterministic or outcome-indeterministic. On the other hand, for a given class of measure-ments to be considered outcome-indeterministic, it is necessaryfor every measurement in the class to be outcome-indeterministic.Consequently, we might have neither outcome-determinism noroutcome-indeterminism for a given class.
Theorem 3
In order to reproduce certain simple fea-tures of operational quantum theory (specifically, the fea-tures P and P outlined in Sec. IV) any ontologicalmodel that is noncontextual in the sense of definition 1must be such that a measurement is represented outcome-deterministically if and only if it is sharp. The proof of this theorem, as well as a discussion of itssignficance, is provided in Sec. IV. First, however, weconsider the question raised in the introduction concern-ing whether considerations parallel to Fine’s theorem canjustify ODUM.
III. THE INTUITION IN FAVOUR OF ODUMAND WHY IT IS MISTAKEN
The most compelling (albeit incorrect) argument infavour of ODUM has the following structure.Premiss 1: If two measurements are statis-tically indistinguishable for all preparations,then they are represented in the ontologicalmodel by the same set of response functions(this is the assumption of measurement non-contextuality).Premiss 2: Every measurement representedin the ontological model by an outcome-indeterministic set of response functions onthe system can also be represented by anoutcome-deterministic set of response func-tions on a composite of the system and anancilla.Purported conclusion: If two measure-ments are statistically indistinguishable forall preparations, then they can be repre-sented in the ontological model by the sameoutcome-deterministic set of response func-tions.The flaw in the argument is not in the premisses; ifthese are clarified appropriately, they are correct. It isthat the conclusion does not follow from the premisses.To get a feeling for why one ought to be suspicious ofthe implication, consider the following analogous argu-ment.Premiss: If two measurements are statisti-cally indistinguishable for all preparations,then they are represented in quantum theoryby the same POVM.Premiss: Every measurement represented inquantum theory by a POVM on the sys-tem can be represented by a projector-valuedmeasure on a composite of the system and anancilla (this is Naimark’s theorem [18]).Purported conclusion: If two measure-ments are statistically indistinguishable for all preparations, then they can be representedin quantum theory by the same projector-valued measure.Here it is clear that the two premisses are true but theconclusion is false. Two measurements procedures on sys-tem+ancilla might be represented by different projector-valued measures, while they nonetheless reduce to thesame POVM on the system and are therefore statisti-cally indistinguishable relative to all preparations of thesystem. This analogy is meant only to give a sense ofwhere the problem lies.In the following, we will make an assumption abouthow the ontological model treats composite systems,namely, that the ontic state space of the composite isthe Cartesian product of the ontic state spaces of thecomponents. For instance, if system s has ontic statespace Λ s and ancilla a has ontic state space Λ a then thecomposite has ontic state space Λ sa = Λ s × Λ a . This as-sumption has been called separability [19] or kinematicallocality [20].We argue that the premisses and the conclusion of theargument in favour of ODUM should be made precise asfollows:Premiss 1 ′ : If two measurements on a sys-tem s are statistically indistinguishable rela-tive to all preparations on s , then they arerepresented in the ontological model by thesame set of response functions on Λ s (this isthe assumption of measurement noncontextu-ality applied to system s ).Premiss 2 ′ : Every measurement on a system s that is represented in the ontological modelby an outcome-indeterministic set of responsefunctions on Λ s can also be understood as theeffective measurement arising from a prepa-ration of an ancilla a and a measurement onsystem+ancilla sa where the latter is repre-sented by an outcome-deterministic set of re-sponse functions on Λ s × Λ a (the measure-ment on sa is said to be the extension of theone on s ).Purported conclusion: If two measurementson a system s are statistically indistinguish-able relative to all preparations on s , then thetwo measurements on system+ancilla sa thatextend these can be represented ontologicallyby the same outcome-deterministic set of re-sponse function on Λ s × Λ a . An explicit argument for premiss 2 ′ is provided in ap-pendix A. Our task now is to show why the conclusiondoes not follow. It suffices to note the following. Harrigan and Rudolph have also previously discussed ontolog-ical models that assign ontic state spaces to the measurement ap-paratus [21].
Claim: If two measurements on a system s are statistically indistinguishable relativeto all preparations on s , it may nonethe-less be the case that the two measurementson system+ancilla sa that are extensions ofthese are statistically distinguishable for somepreparation on sa. This claim is seen to be true simply by noting thata given POVM can have two distinct Naimark exten-sions, that is, there are multiple choices of projector-valued measure on sa that yield the same POVM on s. Although this fact is well-known, we provide an explicitexample in appendix B. It blocks the purported conclu-sion because if two measurements on system+ancilla sa are statistically distinguishable for some preparation on sa, then they must be represented ontologically by dif-ferent response functions on Λ s × Λ a .The conclusion we draw from all of this is that onecan model a pair of measurements that are statisticallyindistinguishable for all preparations on some system byeither:(i) the same outcome- indeterministic set of responsefunctions on the system,(ii) different outcome- deterministic sets of responsefunctions on system+ancilla.So, it is true that one can always represent a measure-ment by an outcome-deterministic set of response func-tion if one wishes; this is done by incorporating otherdegrees of freedom (for instance those of the apparatus)in one’s description. However, in moving to such a rep-resentation, one loses the warrant to apply the assump-tion of measurement noncontextuality. In particular, al-though two measurements on a system may be repre-sented by the same POVM in quantum theory, this isnot enough to justify (via the assumption of measurementnoncontextuality) that they should be represented by thesame set of response functions on system+ancilla in theontological model. The latter conclusion would only bejustified if the two measurements were represented by thesame projector-valued measure on system+ancilla andthis is generally not the case.We can investigate the possibility of a noncontextualontological model using either of conventions (i) or (ii). Itis useful to consider how one models an unsharp measure-ments on system s , denoted M s , within each convention.Under convention (i), we confine our attention to theontic state space of system s , Λ s , we assume outcomeindeterminism, ξ ( k | λ, M s ) ∈ [0 , ∀ λ ∈ Λ s , (5)and we represent the assumption of measurement non-contextuality asIf p ( k | P s , M s ) = p ( k | P s , M ′ s ) ∀ P s ∈ P s then ξ ( k | λ, M s ) = ξ ( k | λ, M ′ s ) ∀ λ ∈ Λ s . (6) Under convention (ii), on the other hand, we expandthe ontic state space from that of the system to thatof system+ancilla, Λ s × Λ a , such that we can assumeoutcome determinism, ξ ( k | λ s , λ a , M sa ) ∈ { , } ∀ ( λ s , λ a ) ∈ Λ a × Λ a , (7)but in this case we must represent the assumption ofmeasurement noncontextuality asif p ( k | P s , M s ) = p ( k | P s , M ′ s ) ∀ P s ∈ P s ,then X λ a µ ( λ a | P a ) ξ ( k | λ s , λ a , M sa )= X λ a µ ( λ a | P ′ a ) ξ ( k | λ s , λ a , M ′ sa ) ∀ λ s ∈ Λ s . (8)In Eq. (8), the expressions on either side of the equal-ity in the consequent are the effective response func-tions on the system. If we think of Eq. (8) as a con-straint on the outcome-deterministic response functions ξ ( k | λ s , λ a , M sa ) and ξ ( k | λ s , λ a , M ′ sa ), it is not simply aconstraint of equality of those response functions. Themore cumbersome nature of this constraint is the priceto pay for insisting on outcome-deterministic representa-tions.In appendix C we provide an explicit example of how,for a particular pair of measurements on the system, eachmeasurement can be associated with a set of responsefunctions that are outcome-deterministic on an extendedsystem—via the scheme of appendix A—but the two setsof response functions fail to be equivalent.These two approaches to providing a noncontextualontological model are analogous to two approaches tooperational quantum theory known as the “church ofthe larger Hilbert space” and the “church of the smallerHilbert space” . In the former, preparations, measure-ments, and transformations on a system are representedrespectively by pure states, projector-valued measures,and unitary maps on system+ancilla, while in the lat-ter, they are represented by mixed states, POVMs andcompletely positive trace-preserving maps on the systemalone. Just as this choice is conventional—either can dojustice to the experimental statistics—so too is the choiceof whether to posit an ontological model on the systemalone or on system+ancilla. Indeed, it is appropriate torefer to convention (ii) as the “church of the larger on-tic state space” and convention (i) as “the church of thesmaller ontic state space”.Conventions (i) and (ii) encode one and the same no-tion of noncontextuality for unsharp measurements. Thepoint is that this notion is distinct from the one that isrecommended in Refs. [9–15].We shall adopt convention (i), i.e. the church of thesmaller ontic state space, for the rest of the article. Terms coined by John Smolin and Matt Leifer respectively.
IV. PROVING THAT OUTCOMEINDETERMINISM HOLDS FOR ALL AND ONLYUNSHARP MEASUREMENTS
We will now prove Theorem 3, which asserts that, givencertain features of operational quantum theory, it fol-lows that in a noncontextual ontological model (usingthe notion of noncontextuality in definition 1) the setof response functions associated with a measurement isoutcome-deterministic if and only if the measurement issharp, or equivalently, that it is outcome-indeterministicif and only if the measurement is unsharp.We divide the proof into two halves:(a) if a measurement is sharp then it is representedoutcome-deterministically in a noncontextual on-tological model,as well as the converse of this implication, which in itscontrapositive form is(b) if a measurement is unsharp then it is representedoutcome- in deterministically in a noncontextual on-tological model,For the proof of (a) we refer the reader to Sec. VIII.Aof Ref. [7]. As noted there, one needs the generalizednotion of noncontextuality from definition 1 to infer out-come determinism for sharp measurements, specifically,one needs the assumption of preparation noncontextu-ality ; the assumption of measurement noncontextualityalone is insufficient to derive the result.The features of operational quantum theory that areused in the proof of (a) can be summarized as follows (asone can infer from the details of the proof, as describedin Ref. [7]):P1 For every sharp measurement M , and for each out-come k of that measurement, there is a preparationprocedure P M,k that makes that outcome certain tooccur, that is, p ( k | M, P
M,k ′ ) = δ k,k ′ . Furthermore,for any two sharp measurements M and M ′ , theuniform mixture over k of the P M,k and the uni-form mixture over k of the P M ′ ,k are statisticallyindistinguishable relative to all measurements.The proof of (b) is a novel contribution of this article.The only sort of noncontextuality that is relevant here ismeasurement noncontextuality.In the course of the proof, we will make critical useof the following fact: If it can be shown that one par-ticular measurement procedure realizing a given POVMmust be represented outcome-indeterministically, thenin a noncontextual ontological model it follows that all measurement procedures realizing that POVM must berepresented outcome-indeterministically. The reason isthat if the two measurement procedures realize the samePOVM, then according to operational quantum the-ory, they are statistically indistinguishable relative to all preparations, and the assumption of measurement non-contextuality then implies that they must be representedby the same set of response functions.To begin, it is useful to note how post-processing ofmeasurements are represented in an ontological model.These constraints have nothing to do with the assump-tion of noncontextuality. They are constraints on any ontological model, contextual or noncontextual.Suppose a measurement M ′ is defined in terms of post-processing of another measurement M as follows.The procedure M ′ : Implement M and uponobtaining outcome k, sample a random vari-able j from a conditional probability distribu-tion s ( j | k ). Finally, output j as the outcomeof the effective measurement.Now consider how these measurements must be repre-sented in an ontological model. If outcome k of mea-surement M is represented by the response function ξ ( k | λ, M ) , the outcome j of measurement M ′ is repre-sented by the response function ξ ( j | λ, M ′ ) = X k s ( j | k ) ξ ( k | λ, M ) . (9)This just follows from probability theory: we must sumthe probability of all the ways of getting outcome j inmeasurement M ′ . We therefore must take the sum over k of the probability of getting outcome k in measurement M and of then getting outcome j in the sampling, andthis latter probability is simply the product of the prob-ability of getting outcome k in M and the probability ofgetting j given k .Note that coarse-graining of measurement outcomes isa special case of post-processing. For instance, if onewishes to coarse-grain all outcomes k in some subset S to a single outcome j = j , one simply chooses the con-ditional such that s ( j | k ) = 1 for all k ∈ S . In thiscase, ξ ( j | λ, M ′ ) = X k ∈ S ξ ( k | λ, M ) . (10)Therefore coarse-graining of outcomes in the operationaltheory is represented by coarse-graining of the corre-sponding response functions.Now consider the operational equivalence class of mea-surement events that are associated with a particu-lar effect E . Denote the spectral resolution of E by E = P i s i Π i where the Π i are projectors satisfying P i Π i = I . We can use the spectral resolution of E to build up a measurement M ′ that has an outcomethat is in the equivalence class associated with E . First,let M denote a sharp measurement associated with theprojector-valued measure { Π i } . Next, define M ′ to be apost-processing of M wherein the conditional probability s ( j | i ) is chosen so that for some value of j , denoted j , wehave s ( j | i ) = s i . The j outcome of measurement M ′ will then be associated with the effect P i s i Π i , which is E . Because of how post-processing is represented in anontological model (see Eq. (9)), we have ξ ( j | λ, M ′ ) = X i s i ξ ( i | λ, M ) . (11)As emphasized above, the assumption of measurementnoncontextuality then implies that for every measure-ment event associated with effect E , not just the eventcorresponding to the j outcome of measurement M ′ , theresponse function for this event is the same. We denoteit by ξ E ( λ ). Denoting the response function for the op-erational equivalence class of the projector Π i by ξ Π i ( λ ),we infer from Eq. (11) that ξ E ( λ ) = X i s i ξ Π i ( λ ) . (12)We are now in a position to demonstrate that allunsharp measurements must be represented outcome-indeterministically. If a quantum measurement is un-sharp, then at least one of its outcomes is associatedwith an effect that is not a projector. Call this effect E and denote its spectral resolution by E = P i s i Π i asbefore, so that its response function is given by Eq. (12).The fact that E is not a projector implies that one ofits eigenvalues, say s i , is such that 0 < s i <
1. Giventhat { Π i } is itself a POVM, it follows that P i ξ Π i ( λ ) = 1in the ontological model (because for every λ , some out-come of the measurement associated with { Π i } must oc-cur). But s i < P i ξ Π i ( λ ) = 1 together imply that P i s i ξ Π i ( λ ) <
1. Finally, because there exist quantumstates that assign a nonzero probability to the i outcomeof the { Π i } measurement, it follows that the responsefunction associated with i must have nontrivial supporton the ontic state space. Hence, for every λ in this sup-port, we have 0 < P i s i ξ Π i ( λ ) <
1, which implies that0 < ξ E ( λ ) <
1. So, ξ E ( λ ) is outcome-indeterministic.It is straightforward to verify that the features of op-erational quantum theory that are used in the proof of(b) are:P2 For every unsharp measurement M ′ , the measure-ment event associated to any given outcome of M ′ can be realized by a post-processing of some sharpmeasurement M where the post-processing is in-trinsically probabilistic rather than deterministic.In other words, if the outcomes of M are labelled by i , then there exists at least one such value, i , suchthat the conditional probability s i of obtaining thedistinguished outcome of M ′ given that outcome i was obtained for M satisfies 0 < s i < A. The significance of Theorem 3
It is worth commenting on how one should interpretThm. 3 given that it is known that operational quantum theory does not admit of an ontological model that isnoncontextual in the sense of definition 1. In particular,in Ref. [7], it was shown that preparation noncontextu-ality alone implies a contradiction with the predictionsof operational quantum theory. As such, one might won-der what is the use of characterizing how measurementsmust be represented in noncontextual ontological modelsof operational quantum theory, as Thm. 3 does, whenit is known that there are no such models. The an-swer is that Thm. 3 is a tool for devising novel proofs ofthe impossibility of a noncontextual ontological model ofquantum theory, in particular, proofs that make nontriv-ial use of the assumption of measurement noncontextu-ality. Indeed, two examples of such proofs are providedin Sec. VII.There are many benefits to having multiple differentproofs of the impossibility of a noncontextual ontologi-cal model of operational quantum theory. After all, thepoint of studying noncontextual ontological models is notmerely to rule them out, but to get a more complete pic-ture of the sense in which operational quantum theorydiffers from a classical theory.It is perhaps easiest to appreciate this point by con-sidering its analogue in the context of proofs of the in-consistency of operational quantum theory and Bell’s as-sumption of local causality. No one can deny that muchhas been learned by exploring the possible paths thatsuch inferences might take, such as the proof provided byHardy [22] or the proof provided by Greenberger Horneand Zeilinger [23]. Just as the different proofs of Bell’stheorem vary in interesting ways, so too do the differentproofs of the impossibility of a noncontextual ontologicalmodel of operational quantum theory.Also, many of the information-processing advantagesof quantum theory can be proven to be connected tothe impossibility of a noncontextual model. The cryp-tographic task of parity-oblivious multiplexing , a kindof random access code, is an example [24]. It has alsorecently been shown that quantum contextuality is aresource in the magic state distilation model of fault-tolerant quantum computation [25]. Understanding thevarious different logical paths from the assumption ofnoncontextuality to a contradiction is important for de-termining which quantum information-processing tasksmight be powered by contextuality.It is also worth noting that the particular features ofoperational quantum theory that are needed to proveThm. 3, namely P P
2, are not by themselves suf-ficient to derive a contradiction with the assumption ofa noncontextual ontological model. One requires addi-tional features of operational quantum theory to obtainthe contradiction. To see that this is the case, it suf-fices to show that there are subtheories of operationalquantum theory—that is, subsets of the full set of prepa- This question was posed by a referee of this article. rations and measurements—that satisfy P P P P do admit of an ontologicalmodel that is noncontextual in the sense of definition 1.This follows from the fact that the preparations and mea-surements in these subtheories have nonnegative Wignerrepresentations (in the stabilizer subtheory, it is the dis-crete Wigner representation of Gross [27]) and the factthat a nonnegative quasiprobability representation yieldsa noncontextual ontological model, as shown in Ref. [28].It follows from Thm. 3 that in these ontological models,measurements are represented outcome-deterministicallyif and only if they are sharp. By inspecting the noncon-textual ontological models that the Wigner representa-tion defines, one can verify that this is indeed the case. V. CONSTRAINTS ON NONCONTEXTUALPROBABILITY-ASSIGNMENTS OVER EFFECTS
In quantum theory, two measurements are statisticallyindistinguishable relative to all preparations if they arerepresented by the same POVM. The POVMs, therefore,describe the operational equivalence classes of measure-ments. It follows that if a measurement M is representedby the POVM { E k } , then measurement noncontextualityimplies that the set of response functions representing M can be labelled by the POVM alone, i.e. no other detailsof the measurement procedure need to be specified, ξ ( k | λ, M ) = ξ ( k | λ, { E k } ) . (13)In fact, measurement noncontextuality implies a fur- ther simplification, namely, that the k th response func-tion can be labelled by the k th element of the POVMalone, that is, ξ ( k | λ, M ) = ξ E k ( λ ) . (14)The proof is straightforward. Consider a measurement M ′ that is obtained from M by coarse-graining all out-comes k = k . Suppose outcome k of M maps to out-come 0 of M ′ and any outcome k = k of M maps tooutcome 1 of M ′ . It is clear that M ′ is then associ-ated with the two-outcome POVM { E k , I − E k } . Wecan therefore label the response functions of M ′ by thisPOVM, ξ ( j | λ, M ′ ) = ξ ( j | λ, { E k , I − E k } ) . (15)Meanwhile, by the definition of M ′ , ξ ( k | λ, M ) = ξ (0 | λ, M ′ ) , (16)and therefore ξ ( k | λ, M ) = ξ (0 | λ, { E k , I − E k } ) . (17)However, given that the POVM { E k , I − E k } is com-pletely specified by specifying E k , it suffices to label theresponse function by E k , ξ ( k | λ, M ) = ξ E k ( λ ) , (18)which is what we set out to prove.It follows from this analysis that in a noncontextualontological model of operational quantum theory, theresponse function associated with a given outcome of aquantum measurement depends only on the POVM ele-ment associated to that outcome .In the rest of this section, we will show how our pro-posal for how to model unsharp measurements in a non-contextual ontological model can be expressed as a set ofconstraints on probability-assignments to effects, ratherthan in terms of constraints on response functions. Thismanner of expressing the proposal is in some respects eas-ier to grasp and clarifies how it contrasts with alternativeproposals.We begin with the traditional notion of noncontextu-ality, which, as mentioned in the introduction, appliesonly to sharp measurements. In terms of the notionsintroduced here, it is the conjunction of the assump-tion of measurement noncontextuality and the assump-tion of outcome determinism for all sharp measurements.Specifically, if M is a sharp measurement with outcomeslabelled by k and associated with the projector-valuedmeasure { Π k } , then, specializing Eq. (14) to projectors,measurement noncontextuality implies that ξ ( k | λ, M ) = ξ Π k ( λ ) . (19)Meanwhile, outcome determinism for sharp measure-ments implies that ξ Π k ( λ ) ∈ { , } . (20)0The traditional assumption of noncontextuality canalso be expressed in terms of constraints on the 0-1 valua-tion of projectors for a fixed ontic state. It is straightfor-ward to verify that this formulation follows from the onein terms of response functions that we have just given.Let v (Π) denote the value assigned to projector Π by afixed ontic state λ , i.e. v (Π) = ξ Π ( λ ) ∈ { , } . Tradi-tional noncontextuality then asserts that for every onticstate, the following conditions hold:KS1 Each projector Π is assigned a value 0 or 1, v (Π) ∈{ , } ,KS2 For each pair of projectors Π , Π , if Π = Π + Π is also a projector, then v (Π) = v (Π ) + v (Π ),KS3 The identity operator is assigned the value 1, v ( I ) = 1.The second item implies that orthogonal projectors can-not both receive the value 1, and the third implies thatfor any set of projectors that form a resolution of identity,exactly one of them must be assigned the value 1. We now reconsider the question of how to modelunsharp measurements in a noncontextual ontologicalmodel from the perspective of valuations over effects.The proposal of Refs. [9–15] is that all effects, eventhe nonprojective ones, should receive 0-1 valuations fora fixed ontic state and that these should satisfy the sameconstraints as do the 0-1 valuations of projectors:1 Each effect E is assigned a value 0 or 1, v ( E ) ∈{ , } ,2 For each pair of effects E , E , if E = E + E isalso an effect, then v ( E ) = v ( E ) + v ( E ),3 The identity operator is assigned the value 1, v ( I ) = 1.These constraints imply that for every POVM, preciselyone effect must receive the value 1 and the others 0.In our approach, on the other hand, nonprojective ef-fects are not assigned deterministic values, but only prob-abilities. Nonetheless, one can express our proposal in It is tempting to think that constraints KS1-KS3 are the con-tent of the assumption of traditional noncontextuality, but this isinaccurate. Rather, the assumption of measurement noncontex-tuality is a prerequisite to KS1-KS3 making any sense. It is thisassumption that warrants positing a function v that depends only on the projector associated with a measurement outcome. So onceone is discussing the properties of a valuation over projectors, theassumption of noncontextuality has already done its work. KS2then follows from how one must represent coarse-graining in anontological model (given by Eq. (10)), KS3 follows from the factthat for every measurement, the sum of probabilities of all the out-comes must be 1, and KS1 encodes the assumption of outcomedeterminism for sharp measurements. terms of constraints on the probability-assignments to ef-fects. Recall that an effect E is a positive operator sat-isfying 0 ≤ E ≤ I . Let w ( E ) denote the probability as-signed to effect E by a fixed ontic state. The constraintson probability assignments to effects are:NC1 Each effect E is assigned a probability, w ( E ) ∈ [0 , E , E , if E = E + E isalso an effect, then w ( E ) = w ( E ) + w ( E ),NC3 For each effect E and for any s satisfying 0 ≤ s ≤ sE is an effect, then w ( sE ) = sw ( E ).NC4 The identity operator is assigned unit probability, w ( I ) = 1.NC5 w ( E ) ∈ { , } if and only if the effect E is a pro-jector, i.e. E = E .We now demonstrate how these constraints can be jus-tified under the assumption of noncontextuality in thesense of definition 1. To translate from ‘response func-tion’ language to ‘probability assignment’ language, wesimply note that an effect E describes an operationalequivalence class of measurement events and the proba-bility assignment to E for ontic state λ is the value of theassociated response function at λ , that is, w ( E ) = ξ E ( λ ).In particular, if outcome k of measurement M is in theequivalence class of measurement events associated with E , then w ( E ) = ξ ( k | M, λ ).NC1 then follows from the fact that ξ ( k | M, λ ) is aprobability.NC2 is simply a consequence of the representation ofcoarse-graining in an ontological model. The equality E = E + E implies that E and E can be associatedwith two distinct outcomes of a single measurement, andthat the coarse-graining of that pair of outcomes is as-sociated with the effect E . However, if two outcomes ofa measurement are coarse-grained into a single outcome,then the probability for the latter given ontic state λ issimply the sum of the probabilities for each of the formergiven λ (see Eq. (10)).NC3 is a consequence of the representation of post-processing in an ontological model. Suppose the effect E is associated with the measurement event correspondingto implementing the measurement procedure M and ob-taining the outcome k . Now define a measurement proce-dure, M ′ , as a post-processing of M as follows: M ′ yieldsoutcome 0 with probability s if M yields outcome k .Clearly, the effect associated with outcome 0 of measure-ment M ′ is then sE . We have already seen how to repre-sent post-processing in an ontological model (see Eq. (9)).From this, we infer that ξ (0 | M ′ , λ ) = sξ ( k | M, λ ), butgiven that w ( E ) = ξ ( k | M, λ ) and w ( sE ) = ξ (0 | M ′ , λ ),we have that w ( sE ) = sw ( E ).NC4 follows from the fact that the probabilities as-signed to the outcomes of any measurement must sum to1.1Finally, NC5 follows from theorem 3. The only sub-tlety in making this inference is that theorem 3 concernsthe representation of the outcomes of sharp measure-ments and hence the representation of projectors that ap-pear in POVMs all of whose elements are projectors whileNC5 refers to a single measurement event associated witha projector, even if that projector appears in a POVMalongside nonprojective effects. We bridge the gap bynoting that measurement noncontextuality implies thatthe response function associated with a projector is thesame regardless of what measurement that projector isconsidered a part of (as implied by Eq. (19)). Because aprojector is represented outcome-deterministically whenit is part of a sharp measurement (by theorem 3), it mustbe represented outcome-deterministically even when it ispart of an unsharp measurement.It is worth noting a few facts about this set of con-straints.First, NC2, NC4 and NC5 together imply KS1, KS2and KS3, so the usual constraints on how to representprojectors in a noncontextual ontological model are re-covered as special cases of our constraints on how torepresent arbitrary effects in a noncontextual ontologi-cal model.It is also useful to note that the constraints NC1-NC5are equivalent to the following set:NC1 ′ Each effect E is assigned a probability equal to oneof the eigenvalues of E , w ( E ) ∈ spec( E ),NC2 ′ For each pair of effects E , E and for each pair ofreals s , s satisfying 0 ≤ s , s ≤
1, if s E + s E is an effect, then w ( s E + s E ) = s w ( E ) + s w ( E ).Equivalence is easy to prove. It is trivial to see thatNC1, NC4 and NC5 all follow from NC1 ′ , while NC2 andNC3 follow from NC2 ′ . Conversely, NC2 ′ follows triviallyfrom NC2 and NC3. Finally, NC1 ′ is derived from NC1-NC5 using the same logic that is used in the proof oftheorem 3. Any effect E has a spectral resolution of theform E = P i s i Π i (where P i Π i = I ) and therefore, byNC2 ′ , w ( E ) = P i s i w (Π i ). However, NC2, NC4 andNC5 imply that w (Π i ) = 1 for precisely one value of i and is zero otherwise (perhaps the easiest way to see thisis by noting that NC2, NC4 and NC5 imply KS1, KS2and KS3), and this in turn implies that w ( E ) = s i forsome value of i . Hence, w ( E ) ∈ spec( E ).In Sec. VII, we will show how this manner of character-izing the consequences of noncontextuality for ontologicalmodels of quantum theory allows one to simplify existingno-go theorems. Before doing so, however, we make useof this distinction between 0-1 valuations and probabilityassignments to frame our final criticisms of ODUM. VI. A DIALOGUE CONCERNING THE STATUSOF ODUM IN A NONCONTEXTUAL MODEL
The compelling but incorrect argument for ODUMthat we considered in Sec. III is, in our estimation, themost likely reason that many researchers have assumedODUM without question. We hope, therefore, that ourcritique of this argument is sufficient to convince suchresearchers to abandon it. Nonetheless, we have alsoshown in Sec. V that the assumption of noncontextualityin definition 1 explicitly implies the failure of ODUM. Inthis section, we provide a few more arguments againstODUM. We present these as a dialogue between the au-thor and an imaginary proponent of ODUM. This pro-vides a more direct confrontation between our approachand the proposal of Refs. [9–15]. Such arguments mayseem redundant at this point, but given that the ODUMproposal has been revived several times by various au-thors, we feel that it may be prudent to drive a few morenails into its coffin.We begin by repeating the argument against ODUMthat was made in Ref. [7]. author:
Consider a measurement procedure M as-sociated with the POVM { I, I } . For anyquantum state ρ , we have tr( ρ I ) = , there-fore this measurement always has equal prob-ability of producing either of its two out-comes, regardless of the preparation proce-dure. Clearly then, one way of implementingthis measurement is as follows: completely ig-nore the system and just flip a fair coin to de-termine the outcome. Call this measurementprocedure M, and consider how it must berepresented in an ontological model. Becausethe outcome doesn’t depend on the system atall, it follows that regardless of the system’sontic state λ , there is a probability of 1 / { , } where each ele-ment should be thought of as a uniform func-tion over λ of height . The outcomes areclearly not deterministic given λ , so outcomedeterminism fails to hold. proponent of odum: Yes, but there are other measurement procedures associated with thePOVM (cid:8) I, I (cid:9) and one of these may berepresented by an outcome-deterministic re-sponse function. author: The problem with this response is thatmeasurement noncontextuality (in the sense We do not claim that any actual proponent of ODUM wouldmake the arguments that are made by our imaginary proponent.Nonetheless, certain parts of our dialogue are inspired by variousproposals for how to define a notion of noncontextuality for unsharpmeasurements, as we note explicitly throughout. regardless of which particular measure-ment procedure is used to implement it. Ittherefore suffices to find just one measure-ment procedure for the POVM that mustbe represented outcome-indeterministicallyto infer that in a noncontextual ontologi-cal model, all measurement procedures forthis POVM must be represented outcome-indeterministically. proponent of odum:
That may be, but it seemsto me significant that one can find sets of un-sharp measurements, which, unlike the faircoin flip POVM have a nontrivial dependenceon the system, and for which it is impos-sible to find a noncontextual and outcome-deterministic representation. Here’s an ex-ample of three such POVMs (due to M. Naka-mura, who was inspired by a similar exampledue to Cabello, both of which are reported inRef. [10]): { E, ( I − E ) , F, ( I − F ) } , { E, ( I − E ) , G, ( I − G ) } , { F, ( I − F ) , G, ( I − G ) } . (21)ODUM implies that for a given λ every ef-fect must be assigned a value of 0 or 1, whilemeasurement noncontextuality implies thatan effect receives the same value regardlessof where it occurs. The contradiction is ob-tained by noting that only one effect in eachPOVM can be assigned the value 1, imply-ing an odd number of 1s, but every effect ap-pears in two POVMs so that a noncontextualassignment must assign an even number of 1s.Such proofs make use of details of the struc-ture of quantum measurements. Doesn’ttheir existence show that ODUM is an inter-esting assumption? author: No, they don’t. If we assume ODUM,then the fair coin flip POVM is already suf-ficient for deriving the contradiction. Giventhat the two effects in the fair coin flip POVMare the same, measurement noncontextualityrequires that we assign them the same value.But we can’t assign them both the value 0because this would say that neither outcomeoccurs, and we can’t assign them both thevalue 1 because this would say that both out-comes occur. We have our contradiction.So we see that proofs of the type describedabove are unnecessarily complicated: a con-sideration of the fair coin flip POVM { I, I } yields the result immediately. The factthat the contradiction can be obtained by a completely trivial argument speaks againstODUM. proponent of odum: Thinking it over, I’ve re-fined my view on the matter. The problemisn’t with ODUM, the problem is with yourdefinition of measurement noncontextuality.It’s too strong. The proper notion of mea-surement noncontextuality for unsharp mea-surements should demand that equivalent ef-fects are represented by equivalent responsefunctions only when these effects appear indistinct POVMs.
Measurement noncontex-tuality should not require equivalent repre-sentations for equivalent effects if these ap-pear in the same POVM. In other words, weneed only eliminate context-dependence be-tween but not within measurements. author:
In my view, the motivation behind theassumption of measurement noncontextualityis that statistical indistinguishability relativeto all preparations should imply statistical in-distinguishability relative to all ontic states,therefore events that have the same statis-tics for all preparations should be representedequivalently in the ontological model, even ifthey correspond to distinct outcomes of a sin-gle measurement.But in any case, even if we consider your sug-gested modification of the notion of measure-ment noncontextuality, it is still trivial to ob-tain a contradiction. Consider the fair coinflip POVM { E , E } where E = E = I together with another POVM containing I ,say { F , F , F } where F = I and F = p I, F = − p I . According to the notion ofmeasurement noncontextuality that you pro-pose, we must require F to take the samevalue as E , but by the same token we mustalso require F to take the same value as E . This implies that E and E must take thesame value, and so we are back to applyingODUM directly to the fair coin flip POVMand obtaining a contradiction trivially. Grudka and Kurzynski [29] have also criticized the notion ofnoncontextuality used in the Cabello-Nakamura proofs. They ar-gue that in a noncontextual model, one should only assign deter-ministic values to the projectors that appear in a
Naimark exten-sion of the POVM, rather than the POVM elements themselves. Itthen suffices to note that the projector that extends a given effectvaries with the POVM in which that effect appears, and thereforethat a noncontextual model does not assign a unique determinis-tic value to a given effect. In the language of the present article,they argue that a noncontextual and outcome-deterministic value-assignment to projectors on system+ancilla does not imply a non-contextual and outcome-deterministic value-assignment to effectson the system. This attitude is entirely consistent with the viewespoused here. proponent of odum: Fine, given this example, Ipropose that the assumption of measurementnoncontextuality together with ODUM sim-ply cannot be applied to any POVM whereinthe same effect appears twice. author: Ah, but this move won’t help either; un-der this new notion one can still constructa contradiction essentially trivially. Con-sider the POVM { p I, − p I, q I, − q I } , where0 < p, q < p, q = and p = q . Supposethat it is coarse-grained in one of two ways:either one coarse-grains the first pair of out-comes, in which case the resulting POVM is { I, q I, − q I } , or one coarse-grains the lastpair of outcomes, in which case the result-ing POVM is { p I, − p I, I } . Now recall thatcoarse-graining at the operational level is rep-resented by coarse-graining at the ontologicallevel, that is, if v ( E ) is the value assigned toeffect E by an ontic state λ , then v ( E + E ) = v ( E ) + v ( E ) . Assuming ODUM, preciselyone of the four effects in { p I, − p I, q I, − q I } must receive the value 1 , but then uponcoarse-graining, the effect I will receive dif-ferent values depending on whether it is con-sidered in the context of { I, q I, − q I } or of { p I, − p I, I } . For instance, if v (cid:0) p I (cid:1) = 1while v ( − p I ) = v ( q I ) = v ( − q I ) = 0 , thenin the context of the first coarse-graining, v ( I ) = v (cid:0) p I (cid:1) + v (cid:0) − p I (cid:1) = 1, while inthe context of the second coarse-graining, v ( I ) = v (cid:0) q I (cid:1) + v (cid:0) − q I (cid:1) = 0. The sameexample would of course not yield a contra-diction if we did not assume ODUM. proponent of odum: You know, I can avoid allof these problems by adopting the followingnotion of noncontextuality for unsharp mea-surements: one is only required to assignoutcome-deterministic response functions toeffects that cannot appear more than once ina given POVM, that is, effects E satisfying E > I . Given that all of your criticisms ofODUM make use of the effect I , and thisdoes not fulfill the conditions for applicabil-ity of such a notion of noncontextuality, yourcriticisms would no longer apply. author: Isn’t this starting to feel like epicycles toyou? In any case, a good operational notionof noncontextuality should apply to any mea-surement. If some proposed notion of non-contextuality for unsharp measurements ne-cessitates a restriction on the sorts of mea- This proposal was considered by Methot [13]. This notion was considered by Bacon, Toner and Ben-Or [30]and reported by Methot [13]. surement to which it can be applied, thenit hasn’t really addressed the problem thatneeds to be solved. If one wants to be ableto say, of any given experiment, whether itadmits of a noncontextual model or not, thedefinition of noncontextuality must be able tocover all cases. proponent of odum:
Well if you insist on a defi-nition that covers all possible measurements,then I’m just going to bite the bullet: myoriginal idea of assuming ODUM in additionto the general notion of measurement noncon-textuality was right all along and the impos-sibility of such an ontological model is indeedtrivial to demonstrate. If your intuitions sug-gested that such proofs should be nontrivial,well then, these examples only demonstratehow wrong those intuitions were. author:
This is a possible position, but it does nothave much to recommend it. Recall that non-contextuality no-go theorems based on sharpmeasurements, such as the original Kochen-Specker theorem, make critical use of struc-tural differences between the set of quantummeasurements and the set of classical mea-surements. All of the trivial no-go theoremsI’ve provided above are based on POVMswherein every element is proportional to theidentity operator. Every such POVM boilsdown to sampling from a probability distri-bution, yielding outcome statistics that arecompletely independent of the preparationprocedure. But of course a classical opera-tional theory also admits of noisy measure-ments that just correspond to sampling froma probability distribution and yield outcomestatistics that are independent of the prepa-ration procedure. Therefore, these trivial no-go theorems do not rely on any intrinsicallyquantum features of the measurements. proponent of odum:
Maybe what this shows isthat we can obtain a no-go result for noncon-textuality even for classical operational the-ories! author:
I take it to be a point in favour of not assuming ODUM that ontological mod-els of classical operational theories are alwaysfound to be noncontextual in the sense of def-inition 1. The whole point of these kinds ofinvestigations is to identify the ways in whichquantum theories and classical theories dif-fer, so a good notion of noncontextuality isone that can do justice to the difference. proponent of odum:
Well, in the end, it’s just aquestion of semantics what one decides to call“noncontextual”. I want to give the nameto one kind of model, you want to give it toanother.4 author:
I don’t agree that the debate is just aboutsemantics. There are criteria by which wecan judge different proposed generalizationsof the notion of noncontextuality: coherenceand usefulness.This article has sought to demonstrate thatthe generalized notion of measurement non-contextuality proposed in Ref. [7], whereinunsharp measurements are associated withoutcome-indeterministic response functions,is coherent. It reduces to the traditional no-tion of noncontextuality for sharp measure-ments and does justice to the manner inwhich unsharp measurements can be relatedto sharp measurements, for instance, whenone is a post-processing of the other.The proposed notion of noncontextuality alsofulfils the criterion of usefulness insofar as itallows us to clarify the ways in which classi-cal and quantum theories differ. Indeed, cer-tain information-processing tasks can be im-plemented better in operational theories thatdo not admit of an ontological model thatis noncontextual in the sense of definition 1,such as the task of parity-oblivious multiplex-ing described in Ref. [24]. A definition ofnoncontextuality that does not cleanly dis-tinguish between classical operational theo-ries and quantum operational theories is lessuseful than one that does.[
End scene. ] VII. DISCUSSION
This article has focussed on the question of how to for-mulate the notion of noncontextuality for unsharp mea-surements. This informs the question of whether one canrule out a noncontextual model of quantum theory usinga proof that appeals explicitly to unsharp measurements.Although there are several results in the literature thatclaim to have done precisely this [9–13], these have allassumed ODUM. This assumption does not follow froman assumption of noncontextuality, however, and conse-quently in the face of a contradiction, one can choose toabandon ODUM in order to salvage noncontextuality. Inour view, therefore, these proofs do not deserve to becalled no-go theorems for noncontextuality.Nonetheless, it is possible to construct a no-go theo-rem for noncontextuality for a single qubit without theassumption of ODUM: this was done in Sec. V of Ref. [7],where two such proofs are presented, one based on a fi-nite set of measurements on a qubit and another thatis based on a version of Gleason’s theorem for unsharpmeasurements that applies to two-dimensional Hilbertspaces, proven by Busch [9] and by Caves et al. [11].We repeat these proofs here, using the characterizationof noncontextuality in terms of probability assignments to effects (NC1 ′ and NC2 ′ in Sec. V), which simplifiesthem substantially.First, the discrete proof. Consider the trine POVM { Π , Π , Π } , where Π , Π and Π are the rank-1projectors which in the Bloch-sphere representation cor-respond to three vectors in a plane separated from oneanother by angles of 120 ◦ , so that Π + Π + Π = I. (22)The assumption of a noncontextual model (via NC2 ′ ) im-plies that the probability assignment over effects inducedby an ontic state must satisfy w (Π ) + w (Π ) + w (Π ) = w ( I ) . (23)However, the assumption of a noncontextual model (viaNC1 ′ ) also implies that w ( I ) = 1 while each of w (Π ), w (Π ) and w (Π ) must take the value 0 or 1. However,no such assignment in consistent with Eq. (23), so wehave derived a contradiction. Note that this proof makesuse of the distinctive features of the space of quantumeffects.The Gleason-like proof is also straightforward. NC2 ′ implies that any probability assignment w to the effects isconvex-linear in the effects. By a standard argument [9,11], it can therefore be extended to a linear function overHermitian operators and hence represented as the innerproduct with another Hermitian operator. Denoting thisoperator by ρ , and recalling that the Hilbert-Schmidtinner product between Hermitian operators A and B istr( AB ), we have w ( E ) = tr( ρE ). The fact that w ( E ) isrequired to be positive for all E implies that ρ must be apositive operator, and the fact that we require w ( I ) = 1implies that tr( ρ ) = 1, therefore ρ is a density operator.To get a contradiction, it then suffices to note that thereis no density operator that assigns to every effect a valuefrom its spectrum. In particular, there is no quantumstate that assigns values 0 or 1 to every projector.Another single-qubit no-go theorem for noncontextual-ity has been provided recently by Kunjwal and Ghosh [31]who demonstrate a violation of a noncontextuality in-equality derived in Ref. [32] for a triple of nonprojectivequbit POVMs that are jointly measurable pairwise butnot triplewise.One consequence of our analysis is that the restric-tion of previous no-go theorems for noncontextuality toHilbert spaces of dimension 3 or greater was an artifact ofhaving a notion of noncontextuality that was limited tosharp measurements. For a qubit, there is only a singlemeasurement context in which any given rank-1 projec-tor can appear, namely, together with its unique rank-1orthogonal complement. Hence, there is no possibility ofa nontrivial variation of the context in which a projectorappears and hence no possibility of context-dependenceeither. When one considers unsharp measurements, onthe other hand, there are nontrivial contexts: a givennonprojective POVM may be realized as a convex com-bination of other measurements in multiple ways, as a5post-processing of other measurements in multiple ways,and by reduction of another measurement in multipleways. However, for unsharp measurements, achieving anoncontextual model is not about assigning outcomes ina context-independent fashion, it is about assigning prob-abilities of outcomes in a context-independent fashion.Finally, a few words are in order regarding the motiva-tion that we provided for our investigation—determiningwhether experimental statistics can be explained by anoncontextual ontological model, irrespective of the truthof quantum theory. This article has not addressed thisquestion directly; we have only considered the constraintson noncontextual ontological models of quantum theory .The distinction between sharp and unsharp measure-ments, for instance, is defined in terms of quantum con-cepts. What our analysis demonstrates, however, is thatone should never simply assume that the noncontextualrepresentation of some measurement should be outcome-deterministic. Rather, this needs to be justified. As wehave noted, Ref. [7] demonstrated that one can justifyoutcome-deterministic representations of sharp measure-ments in quantum theory from preparation noncontextu-ality and from certain facts about the quantum statistics.In the context of exploring whether given experimen-tal data can be explained by a noncontextual ontologi-cal model, any assumptions of outcome-determinism willsimilarly need to be justified by assumptions of noncon-textuality and by appeal to facts about the experimentalstatistics. Or, more precisely, the degree of indetermin-ism posited for some noncontextual representation of ameasurement needs to be so justified. An example ofa noncontextuality inequality for experimental statisticswherein the degree of indeterminism is justified in thismanner will be provided elsewhere [33]. Acknowledgments
The author would like to thank John Sipe and HowardWiseman for their insistence that the topic of outcome-determinism for unsharp measurements deserved a bettertreatment. Thanks also to Ernesto Galv˜ao, Ben Toner,Howard Wiseman, and Ravi Kunjwal for discussions, andto an anonymous referee for suggesting a simplification ofthe proof of Thm. 3. Research at Perimeter Institute issupported in part by the Government of Canada throughNSERC and by the Province of Ontario through MRI.
Appendix A: Ontological extension: modelling anunsharp measurement using anoutcome-deterministic response function
The intuition at play in premiss 2 of the argumentfrom Sec. III is that one can always imagine any sub-jective uncertainty in the outcome of a measurement onsome system s as being due to uncertainty about theontic state of some other system that, together with s, determines the outcome, for instance, hidden variablesin the apparatus. We will see that this is indeed thecase. The resulting representation of the measurementwill be called an ontological extension . This is the ana-logue, within the ontological model, of the Naimark ex-tension of a measurement within operational quantumtheory. Suppose that for an operational theory ( P , M , p ) onsystem s, we have found a (possibly contextual) onto-logical model (Λ s , µ, ξ ) on s, such that the operationalstatistics are reproduced as p ( k | M, P ) = X λ s ∈ Λ s µ ( λ s | P ) ξ ( k | λ s , M ) . (A1)We suppose that the ontological model is outcome-indeterministic, so that 0 < ξ ( k | λ s , M ) < λ s . We can define a new model with outcome-deterministicresponse functions as follows. Introduce an ancilla sys-tem a with ontic state space equal to the unit interval(hence continuous), that is, Λ a = [0 , sa ≡ Λ s × Λ a . Now define µ ( λ s , λ a ) ≡ µ ( λ s | P ) µ ( λ a ) , (A2)where µ ( λ a ) is the uniform distribution over [0 , ω ( λ s ) = 0 , (A3) ω k ( λ s ) = k X j =1 ξ ( j | λ s , M ) , (A4)and define outcome-deterministic response functions onΛ s × Λ a by ξ ( k | λ s , λ a ) ≡ (cid:26) ω k − ( λ s ) ≤ λ a ≤ ω k ( λ s )0 otherwise . (A5) Just as one can define a quantum Naimark extension by adjoin-ing an ancilla to one’s system or by considering the system to bea subspace of a larger system, so too can one define an ontologicalNaimark extension in either way. We’ll use the ancilla constructionhere. It is possible that one could dispense with the assumption ofseparability if one used the subspace construction, but we do notseek to answer the question here. X λ s ∈ Λ s Z Λ a d λ a µ ( λ s , λ a ) ξ ( k | λ s , λ a ) (A6)= X λ s ∈ Λ s µ ( λ s | P ) Z ω k ( λ s ) ω k − ( λ s ) d λ a µ ( λ a ) (A7)= X λ s ∈ Λ s µ ( λ s | P ) [ ω k ( λ s ) − ω k − ( λ s )] (A8)= X λ s ∈ Λ s µ ( λ s | P ) ξ ( k | λ s , M ) (A9)= p ( k | P, M ) . (A10)A similar trick for devising an ontological model thatis outcome-deterministic was described by Bell [1] andits relevance for eliminating determinism as an assump-tion in the proof of Bell’s theorem was emphasized byFine [16]. It follows that one can always extend anontological model such that it represents measurementsoutcome-deterministically if one so wishes. This justifiespremiss 2 ′ of the second argument in Sec. III. Appendix B: The multiplicity of Naimark extensionsof a POVM and its significance for ontologicalmodels
We here show explicitly that a given POVM can havetwo distinct Naimark extensions, that is, that there aremultiple distinct choices of projective measurement on sa that yield the same POVM on s. Specifically, we con-struct two Naimark extensions of the fair coin flip POVM { I, I } (discussed in Sec. VI).The first Naimark extension is as follows. We imple-ment a preparation of the ancilla corresponding to themixed state ρ a = | i h | + | i h | + | i h | . The measurement on sa is the binary-outcome projector-valued measure { Π sa , I − Π sa } , whereΠ sa ≡ Π s, ⊗ | i h | + Π s, ⊗ | i h | + Π s, ⊗ | i h | . (B1)where Π s,i ≡ | ψ i i h ψ i | and {| ψ i i : i ∈ { , , }} are atriple pure states on the system s with pairwise overlaps |h ψ i | ψ j i| = for i = j , that is, in the Bloch representa-tion, they are separated by 120 ◦ on an equatorial plane ofthe Bloch sphere. To see that the effective measurementon the system is the POVM { I, I } , it suffices to notethat Tr a (Π sa ρ a ) = Π s, + Π s, + Π s, = I. The second Naimark extension is as follows. We im-plement a preparation of the ancilla corresponding to themixed state σ a = | i h | + | i h | , (B2) and implement a measurement on the composite ofsystem+ancilla corresponding to the three-outcomeprojector-valued measure { I ⊗ | i h | , I ⊗ | i h | , I ⊗ | i h |} (B3) ≡ { Π sa, , Π sa, , Π sa, } (B4)(although technically this is a measurement on the com-posite, it is obviously only nontrivial on the ancilla).Again, it is straightforward to verify that the effectivemeasurement on the system is the POVM { I, I } .The pair of projector-valued measures appearingin the two Naimark extensions, { Π sa , I − Π sa } and { Π sa, , Π sa, , Π sa, } , are clearly distinct. This impliesthat there are quantum states on sa that yield differentstatistics for the two. For instance, the state I ⊗ | i h | yields 50/50 statistics for { Π sa , I − Π sa } but alwaysyields the third outcome for { Π sa, , Π sa, , Π sa, } . Thisimplies that although these projector-valued measuresare each represented by a set of outcome-deterministic response functions in an ontological model that is non-contextual in the sense of definition 1 (by virtue of the-orem 3), nonetheless the two sets of response functions must be different to account for the differing statistics. Appendix C: An explicit example of ontologicalextension
We have seen in appendix B that because a quan-tum measurement can be Naimark-extended in manyways, and because those extensions may be statisti-cally distinguishable, it follows that the sets of re-sponse functions that represent these Naimark exten-sions must also be inequivalent. This section providesa second way of understanding the fact that the costof modelling statistically-indistinguishable measurementsoutcome-deterministically is that on the extended sys-tem, their representations are no longer equivalent. Weconsider the trick described in appendix A for replac-ing a response function on the system with an outcome-deterministic response function on an extension of thesystem. Specifically, we show by example that a pair ofmeasurements that are represented by the same set ofresponse functions on the system may need to be repre-sented by different sets of response functions on its ex-tension.Let M be a measurement associated with the fair coinflip POVM { I, I } that was discussed in Sec. VI. Letthe second measurement M ′ be defined as follows.The procedure M ′ : implement M , and uponobtaining outcome b ∈ { , } , output b ⊕ b .Equivalently, we can say simply that M ′ is a measure-ment of M with the outcomes permuted. Clearly M ′ isalso represented by the fair coin flip POVM. Now con-sider how to represent M and M ′ within a noncontextual7ontological model. As argued in Sec. VI, the set of re-sponse functions must be simply { , } . By the construction in appendix A, we can model both M and M ′ on a larger system, a composite of system s and an ancilla a with ontic state space Λ s × Λ a , insuch a way that the response functions are outcome-deterministic. Specifically, if we apply this constructionto M , we obtain ξ (0 | λ s , λ a , M ) ≡ (cid:26) ≤ λ a ≤ /
20 otherwise , (C6) ξ (1 | λ s , λ a , M ) ≡ (cid:26) / ≤ λ a ≤
10 otherwise . (C7)Meanwhile, if we remember the definition of M ′ , it is clear that it must be represented by a set of responsefunctions that is simply the permutation of those for M , ξ (0 | λ s , λ a , M ′ ) = ξ (1 | λ s , λ a , M ) , (C8) ξ (1 | λ s , λ a , M ′ ) = ξ (0 | λ s , λ a , M ) . (C9)Now we see that although we’ve managed to have both M and M ′ represented by sets of response functions onΛ s × Λ a that are outcome-deterministic, the responsefunctions are not equivalent, ξ (0 | λ s , λ a , M ′ ) = ξ (0 | λ s , λ a , M ) , (C10) ξ (1 | λ s , λ a , M ′ ) = ξ (1 | λ s , λ a , M ) . (C11) [1] J. S. Bell, “On the problem of hidden variables inquantum mechanics”, Rev. Mod. Phys. , 447 (1966);Reprinted in Ref. [3], chap. 1.[2] S. Kochen and E. P. Specker, “The Problem of hiddenvariables in quantum mechanics’, J. Math. Mech. , 59(1967).[3] J. S. Bell, Speakable and unspeakable in quantum mechan-ics (Cambridge University Press, New York, 1987).[4] A. Cabello and G. Garcia-Alcaine, “Proposed experi-mental tests of the Bell-Kochen-Specker theorem”, Phys.Rev. Lett. , 799-805 (2002).[7] R. W. Spekkens, “Contextuality for preparations, trans-formations, and unsharp measurements”, Phys. Rev. A , 052108 (2005).[8] J. S. Bell, “On the Einstein-Podolsky-Rosen Paradox”,Physics , 195 (1964). Reprinted in Ref. [3], chap. 2.[9] P. Busch, “Quantum states and generalized observables:a simple proof of Gleason’s theorem”, Phys. Rev. Lett. , 120403 (2003).[10] A. Cabello, “Kochen-Specker Theorem for a Single Qubitusing Positive Operator-Valued Measures”, Phys. Rev.Lett. , 190401 (2003).[11] C. M. Caves, C. A. Fuchs, K. Manne, and J. M. Renes,“Gleason-Type Derivations of the Quantum ProbabilityRule for Generalized Measurements,” Found. Phys. ,193 (2004).[12] P. K. Aravind, “The generalized Kochen-Specker theo-rem”, Phys. Rev. A , 052104 (2003)[13] A. A. Methot, “Minimal Bell-Kochen-Specker proofswith POVMs on qubits”, Int. J. Quantum Inf. , 353(2007).[14] Q. Zhang, H. Li, T. Yang, J. Yin, J. Du, J. W. Pan,“Experimental Test of the Kochen-Specker Theorem forSingle Qubits using Positive Operator-Valued Measures”,arXiv preprint quant-ph/0412049 (2004).[15] L. Mancinska, G. Scarpa, and S. Severini. “New Separa-tions in Zero-Error Channel Capacity Through Projec-tive KochenSpecker Sets and Quantum Coloring,” IEEE Transactions on Information Theory , 4025 (2013).[16] A. Fine,“Hidden variables, joint probability, and the Bellinequalities,” Phys. Rev. Lett. , 291 (1982).[17] C. Simon, C. Brukner and A. Zeilinger, “Hidden-variabletheorems for real experiments”, Phys. Rev. Lett. , 4427(2001).[18] A. Peres, Quantum Theory: Concepts and Methods (Kluwer Academic, Boston, 1995).[19] N. Harrigan and R. W. Spekkens,“Einstein, incomplete-ness, and the epistemic view of quantum states,” Found.Phys. , 125 (2010).[20] R. W. Spekkens, “The paradigm of kinematics and dy-namics must yield to causal structure,” arXiv preprintquant-ph/1209.0023 (2012).[21] N. Harrigan and T. Rudolph. “Ontological models andthe interpretation of contextuality,” arXiv:0709.4266(2007).[22] L. Hardy, “Nonlocality for two particles without inequal-ities for almost all entangled states.” Phys. Rev. Lett. , 1665 (1993).[23] D. M. Greenberger, M. A. Horne, A. Zeilinger, “Goingbeyond Bells theorem”. In Bells theorem, quantum the-ory and conceptions of the universe (pp. 69-72). SpringerNetherlands (1989).[24] R. W. Spekkens, D. H. Buzacott, A. J. Keehn, B. Toner,G. J. Pryde, “Preparation Contextuality Powers Parity-Oblivious Multiplexing”, Phys. Rev. Lett. , 010401(2009).[25] M. Howard, J. Wallman, V. Veitch, and J. Emerson,”Contextuality supplies the magic for quantum compu-tation”. Nature
351 (2014).[26] S. D. Bartlett, T. Rudolph, R. W. Spekkens, “Recon-struction of Gaussian quantum mechanics from Liouvillemechanics with an epistemic restriction”. Phys. Rev. A , 012103 (2012).[27] D. Gross, “Hudson’s theorem for finite-dimensional quan-tum systems”. J. Math. Phys. ,020401 (2008).[29] A. Grudka and P. Kurzynski, “Is there contextuality fora single qubit?”, Phys. Rev. Lett. , 160401 (2008).[30] B. F. Toner, D. Bacon and M. Ben-Or, “Kochen-Specker theorem for Generalized Measurements”, unpublishedmanuscript (2005).[31] R. Kunjwal, S. Ghosh, “A minimal state-dependent proofof measurement contextuality for a qubit,” Phys. Rev. A , 042118 (2014).[32] Y. C. Liang, R. W. Spekkens, H. M. Wiseman, “Specker’s parable of the overprotective seer: A road to contextual-ity, nonlocality and complementarity”. Phys. Rep.506