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Existential Contextuality and the Models of Meyer, Kent and Clifton

It is shown that the models recently proposed by Meyer, Kent and Clifton (MKC) exhibit a novel kind of contextuality, which we term existential contextuality. In this phenomenon it is not simply the pre-existing value but the actual existence of an observable which is context dependent. This result confirms the point made elsewhere, that the MKC models do not, as the authors claim, ``nullify'' the Kochen-Specker theorem. It may also be of some independent interest.


I. INTRODUCTION
Meyer [1], Kent [2] and Clifton and Kent [3] (MKC in the sequel) have recently proposed a new class of hidden variables models in which values are only assigned to a restricted subset of the set of all observables. MKC claim that their models "nullify" the Kochen-Specker theorem [4,5,6,7,8]. In Appleby [9] we showed that this claim is unfounded: the MKC models do not, in any way, invalidate the essential physical point of the Kochen-Specker theorem (for other critical discussions see refs. [11,12,13,14,15]). Nevertheless, the MKC models are still of much interest. Together with the models proposed by Pitowsky [16] they show that the physical interpretation of the Kochen-Specker theorem involves some important subtleties which, in the past, have not been sufficiently appreciated.
The purpose of this paper is to show that the MKC models exhibit a novel kind of contextuality, which has not previously been remarked in the literature, and which is even more strikingly at variance with classical assumptions than the usual kind of contextuality, featuring in the Kochen-Specker theorem. In the usual kind of contextuality it is only the value assigned to an observable which is context-dependent. In the MKC models, however, it is the very existence of an observable which is context-dependent (its existence, that is, as a physical property whose value can be revealed by measurement). This phenomenon may be described as existential contextuality [29]. It confirms the point made in ref. [9], that the MKC models do not, as MKC claim, provide a classical explanation for non-relativistic quantum mechanics.
This paper was originally motivated by a seeming inconsistency in MKC's statement [2,3], that their models are both non-contextual and non-local. There do, of course, exist theories which have both these properties (Newtonian gravity, for example). However, in the framework of quantum mechanics the phenomena of contextuality and nonlocality are closely connected, as has been stressed by Mermin [6] (also see Heywood and Redhead [17] and Basu et al [10]). The discussion in Mermin [6] suggests that, if one were to examine the predictions the MKC models make regarding (for example) the GHZ set up [20], then one might expect to find evidence that these models are not only non-local (as MKC state), but also contextual (which they deny). As we will see, this is in fact the case.

II. GHZ SET-UP, WITH A LOCALITY ASSUMPTION
Consider Mermin's variant [6,21] of the GHZ set-up [20], as illustrated in Fig. 1. This arrangement is usually regarded as a way of demonstrating non-locality. As discussed in the Introduction, we will show that it can also be used to demonstrate a form of contextuality.
The system consists of three spin-1/2 particles. Letσ (r) denote the Pauli spin vector for particle r, and let H (r) be the 2-dimensional Hilbert space on which it acts. The spin state is where |s 1 , s 2 , s 3 denotes the joint eigenstate ofσ (1) z ,σ z ,σ (3) z with eigenvalues s 1 , s 2 , s 3 . The particles emerge from a source and pass through three space-like separated detectors (see Fig. 1). For each r the corresponding detector measures one of the two observablesσ (r) x orσ (r) y . One has [21] σ (1) Consequently, if the detectors are strictly ideal, and if they are set precisely at the combination xxx, then the product of measured values must necessarily be −1. Similarly, if the detectors are strictly ideal, and if they are set precisely at one of the combinations xyy, yxy, yyx, then the product of measured values must necessarily be +1. MKC argue that it would not, in practice, be possible to align the detectors with infinite precision, implying that the detectors, instead of performing ideal measurements [30] of the observablesσ j3 (with j r = x or y), may actually perform ideal measurements of a slightly different set of commuting observablesτ (1) ,τ (2) ,τ (3) . They postulate that the observablesτ (1) ,τ (2) ,τ (3) are always such that their joint spectral resolution is a subset of a countable set P d (in the notation of Clifton and Kent [3]), which is dense in the space of all projections on H 1 ⊗ H 2 ⊗ H 3 . For each r detector r reveals the pre-existing value of the observableτ (r) which it does in fact ideally measure.
The fact that the observablesτ (r) may not precisely coincide with the observablesσ (r) jr means that there may be a small, non-zero probability of obtaining the "wrong" measurement outcome (i.e. 1 for the combination xxx, and −1 for the combinations xyy, yxy, yyx). This is consistent with the unavoidable imprecision of real, laboratory measurements.
In this paper we are, for simplicity, confining ourselves to the kind of measurement envisaged by MKC, in which the imprecision is entirely due to the detectors not being aligned precisely in the directions specified. It should be stressed that such measurements are still highly idealised. MKC assume that there is always some observable which a detector ideally measures. They overlook the fact that a real, laboratory instrument does not, typically, perform an ideal measurement of anything: neither the nominal observable, which the experimenter records as having been measured, nor any other observable either. We discuss this point further in Appleby [9].
In their published papers MKC take the view that the difference betweenτ (r) andσ (r) jr is due to detector r not being aligned with infinite precision. On this viewτ (r) must be a local observable of the formτ (r) = n r ·σ (r) , where n r is a unit vector close to the unit vector in the j r direction, representing the actual alignment [31] of detector r. Kent's [22] subsequent suggestion, thatτ (r) may be a non-local admixture of observables pertaining to more than one particle, will be discussed in the next section.
Given an arbitrary triplet of unit vectors (n 1 , n 2 , n 3 ), define projectionŝ where, for each r, s r = ±1. These projections constitute the joint spectral resolution for the operators n 1 ·σ (1) , n 2 ·σ (2) , n 3 ·σ (3) . Define S ′ 6 to be the set of vector triplets (n 1 , n 2 , n 3 ) for which the corresponding projectionsP s1s2s3 all ∈ P d (where, in the notation of Clifton and Kent [3], P d is the countable set of projections on which the MKC valuations are defined). S ′ 6 is a countable, dense subset of S 2 × S 2 × S 2 (where S 2 is the unit 2-sphere). Its significance is that (n 1 , n 2 , n 3 ) represents a possible set of alignments for the three detectors if and only if (n 1 , n 2 , n 3 ) ∈ S ′ 6 . We will now show that S ′ 6 cannot be a Cartesian product of the form We will then use this to show that the MKC models exhibit a novel kind of contextuality.
In order to establish this result suppose that S ′ 6 is of the form We will show that this assumption leads to a contradiction.
For each r let n rx , n ry be a fixed pair of vectors ∈ S ′ (r) 2 such that n rx (respectively n ry ) is close to e x (respectvely e y ), the unit vector in the x (respectively y) direction. Then where ǫ a ≥ 0 for each a. Let ǫ = max(ǫ 0 , ǫ 1 , ǫ 2 , ǫ 3 ). It follows from Eqs. (2) and (3) and the continuity of the expectation values that ǫ → 0 as n rx → e x , n ry → e y for r = 1, 2, 3. The fact that S ′ 6 is dense in S 2 × S 2 × S 2 means that S ′ (r) 2 is dense in S 2 for r = 1, 2, 3. It follows that the vectors n rj can be chosen so as to make ǫ arbitrarily small. Let Λ be the hidden state space, and for each λ ∈ Λ let s rj (λ) be the corresponding valuation of n rj ·σ (r) . We have s rj (λ) = ±1 for all r, j. Define Then f a (λ) = ±1 for all a, λ. Also for all λ.
Let µ be the probability measure on Λ corresponding to the state |ψ . The assumption that S ′ for all a. For each a let A a be the set Then it follows from Inequality (8) that for all a. It follows that µ(A a ) ≥ 1 − ǫ/2 for all a and, consequently, that µ( We noted above that, with a suitable choice of the vectors n rj , ǫ can be made arbitrarily small. It follows that there exist vectors n rj such that µ( On the other hand, it follows from Eq. (7) that µ(A 0 ∩ A 1 ∩ A 2 ∩ A 3 ) = 0 for every choice of n rj -which is a contradiction.
We have thus shown that the set S ′ 6 does not have the form of a Cartesian product for any model of MKC type. This has important consequences: for it implies that it must, in general, happen that a change in the alignment of one detector forces a change in the alignment of at least one of the other two detectors. This represents a form of non-locality. However, the point which concerns us here is that it also represents a form of contextuality.
It is a particularly striking form of contextuality. Let S ′ (r) 2 be the set of possible alignments for detector r. In the usual kind of contextuality S ′ (r) 2 is fixed, and it is only the values assigned to the members of this set which depend on the measurement context. However, in the MKC models it is the set S ′ (r) 2 itself which depends on the measurement context. In other words, it is not simply the value, but the very existence of an observable which is context-dependent (its existence, that is, as a physical property whose value can be revealed by measurement).

III. GHZ SET-UP, WITH NON-LOCAL DETECTORS
In the last section we assumed that detector r reveals the value of a local observable, defined on the state space of particle r. Kent [22] has objected to this assumption. He suggests, instead, that, on the level of the hidden variables, the detectors may function as non-local devices, which reveal the values of non-local admixtures of observables pertaining to more than one particle.
Let us begin by noting that this suggestion involves a significant departure from the view taken in MKC's published papers. In their published papers MKC argue that the observableτ (r) , whose value is revealed by detector r, is also the observable which detector r ideally measures [32], the discrepancy betweenτ (r) andσ (r) jr being entirely attributable to an inaccuracy in the alignment of detector r. Clearly, detector r can only perform ideal quantum measurements of local observables pertaining to particle r. Consequently, the position adopted in MKC's published papers implies thatτ (r) must be a local observable pertaining to particle r-as we assumed in the last section.
If a detector reveals the pre-existing value of some non-local observable then, on the level of the hidden variables, it must be interacting non-locally with more than one particle. This interaction would represent a further element of non-locality in the theory, additional to the non-locality required by the standard arguments (Bell, GHZ, etc.). A model of this kind would thus be even more strongly non-classical than the models originally proposed in MKC's published papers.
Nevertheless, the fact that the observablesτ (r) may be assumed to be arbitrarily close to local observables of the form n r ·σ (r) means that a model of the kind indicated will still be consistent with the empirical predictions of conventional quantum mechanics. The question consequently arises, whether the phenomenon of existential contextuality, discussed in the last section, also occurs in models of this more general kind. It is easily seen that the answer to this question is in the affirmative.
Let P be the set of all projection operators on H 1 ⊗ H 2 ⊗ H 3 , and let P d be the countable, dense subset of P on which the MKC truth-functions are defined (where we are employing the notation of Clifton and Kent [3], as before). LetP d be the set of self-adjoint operators on H 1 ⊗ H 2 ⊗ H 3 whose spectral resolutions are contained in P d . The triplet (τ (1) ,τ (2) ,τ (3) ) of commuting observables whose values are revealed by the three detectors must ∈P d ×P d ×P d . It is determined by the hidden state of the three detectors. Let T ⊆P d ×P d ×P d be the set of all possible triplets (τ (1) ,τ (2) ,τ (3) ), as determined by the set of all possible hidden detector states. The set T is the analogue, in the more general setting of this section, of the set S ′ 6 defined in the last section. We may now show, using a straightforward modification of the argument in the last section, that T is not a Cartesian product. In fact, suppose that T was of the form T (1) × T (2) × T (3) (with T (r) ⊆P d for r = 1, 2, 3). We could then choose, for each r = 1, 2, 3 and j = x, y, operatorsτ (r) j ∈ T (r) such thatτ (r) j ≈σ (r) j for all r, j. This would imply where the positive constant ǫ can be chosen arbitrarily small (compare Eqs. (5) in the last section). If ǫ < 1/2 we can show that these inequalities lead to a contradiction, by an argument which is essentially the same as the argument following Eqs. (5) in the last section. It follows that T is not a Cartesian product. We conclude that the MKC models still exhibit the phenomenon of existential contextuality described in the last section, even on the assumption that the detectors may reveal the pre-existing values of non-local observables.

IV. CONCLUSION
In this paper we have argued that the MKC models are contextual. It follows that they do not, as MKC claim, provide a classical explanation for the empirically verifiable predictions of non-relativistic quantum mechanics. This confirms the conclusion reached in Appleby [9], on the basis of a different, completely independent argument.
We would, however, stress that, notwithstanding these criticisms, it appears to us that the work of MKC is deeply interesting, and important. We have argued that MKC's attempt to explain non-relativistic quantum mechanics in classical terms is misconceived. Nevertheless their work is still valuable because, together with the earlier work of Pitowsky [16], it shows that the physical interpretation of the Kochen-Specker theorem is a great deal more subtle than may superficially appear. It consequently leads to a deeper understanding of the conceptual implications of quantum mechanics.
The work of MKC and Pitowsky is also interesting because it enlarges the scope of the hidden variables concept in a new and imaginative way. In the past hidden variables theories have primarily been motivated by purely philosophical considerations. The emphasis has been largely (though not entirely-see Valentini [23,24]) on constructing alternative interpretations of conventional quantum mechanics. Recently, however, 't Hooft [25,26] (in one way) and Faraggi and Matone [27] and Bertoldi et al [28] (in another way) have speculated that Planck scale physics may most appropriately be described in terms of a hidden variables theory which is not equivalent to conventional quantum mechanics. A theory of this kind, if it could be constructed, would be empirically significant.
In this connection it may be worth noting that 't Hooft [26] has argued that, on the level of Planck scale variables, it may not be possible to rotate a detector at will so as to measure either the x or y components of a particle's spin, and that this may provide a way of circumventing the Bell theorem. This proposal is similar to MKC's attempt to circumvent the Kochen-Specker theorem. Our analysis of the MKC models would consequently seem to indicate that one cannot restore classicality in the manner 't Hooft suggests.
On the other hand, there is no evident reason why one should demand non-contextuality and locality in respect of a theory of the kind proposed by 't Hooft. Such a theory must, by definition, restore the concept of a world of objective facts. However, this concept is by no means exclusive to classical physics. In other respects the theory might be highly non-classical. Indeed, it might be even more highly non-classical than conventional quantum mechanics. The aim is to understand the actual constitution of the physical universe. There is no clear reason to exclude, at the outset, the possibility that the world actually is contextual and non-local.
The ideas of 't Hooft and Faraggi et al are admittedly speculative. They do, however, provide an additional motive for investigating new and more imaginative implementations of the hidden variables hypothesis.