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The Question of Empirical Testing

Famously, the violation of Bell's inequalities, prescribed by QM, has been confirmed experimentally. Is something similar possible for the KS theorem? We shoulddistinguish three questions: (1) Is it possible to realize the experiment proposed by KS as a motivation of their theorem? (2) Is it possible to test the principles leading to the theorem: the Sum Rule and Product Rule, FUNC, or NC? (3) Isit possible to test the theorem itself?

(1) KS themselves describe a concrete experimental arrangement to measure Sx2, Sy2, Sz2 on a one-particle spin-1 system as functions of one maximal observable.An orthohelium atom in the lowest triplet state is placed in a small electric field E of rhombic symmetry. The three observables in question then can be measured as functions of one single observable, the perturbation Hamiltonian Hs. Hs, bythe geometry of E, has three distinct possible values, measurement of which reveals which two of Sx2, Sy2, Sz2 have value 1 and which one has value 0 (see Kochen and Specker 1967: 72/311). This is, of course, a proposal to realize an experiment exemplifying our above value constraint (VC2). Could we also realize a (VC1) experiment, i.e. measure a set of commuting projectors projecting on eigenstates of one maximal observable? Peres (1995: 200) answers the question in the affirmative, discusses such an experiment, and refers to Swift and Wright (1980) for details about the technical feasibility. Kochen and Specker's experimental proposal has, however, not been further pursued, because it does not provide a direct test of NC. Obviously, a measurement of HS measures one orthogonal triple onl

y. An HV proponent might well assume that the hidden state changes from one measurement of HS to the next (even if we prepare the same QM state again) and thusmaintain NC.

(2) In conjunction with manifestations of FUNC, i.e. the Sum Rule and the Product Rule, QM yields constraints like VC1 or VC2 that contradict VD. So, providingconcrete physical examples that could, given the Sum Rule and the Product Rule,instantiate VC1 or VC2 as just outlined is not enough. We must ask whether theserules themselves can be empirically supported. There was considerable discussion of this question in the early 80s explicitly about whether the Sum Rule is empirically testable and there was general agreement that it is not.[15]

The reason is the following. Recall that the derivation of FUNC established uniq

ueness of the new observable f(Q) only in its final step (via NC). It is this uniqueness which guarantees that one operator represents exactly one observable sothat observables (and thereby their values) in different contexts can be equated. This allows one to establish indirect connections between different incompatible observables. Without this final step, FUNC must be viewed as holding relative to different contexts, the connection is broken and FUNC is restricted to oneset of observables which are all mutually compatible. Then indeed FUNC, the SumRule and the Product Rule become trivial, and empirical testing in these cases would be a pointless question.[16] It is NC that does all the work and deserves to be tested via checking whether for incompatible P, Q such that f(Q)=g(P) it istrue that v(f(Q))=v(g(P)). However, though QM and a noncontextual HV theory contradict each other for a single system, this contradiction involves incompatibleobservables and, thus, is untestable (as we have just seen from Kochen and Spec

ker's own proposal). Physicists have, however, made ingenious proposals for overcoming this obstacle. It is well-known that the consideration of two-particle systems and products of spin components leads to very simple KS-type proofs (Mermin 1990b). Cabello and Garca-Alcaine (1998) have shown that for such systems QM and a noncontextual HV theory make different predictions for every single case. Their reasoning makes no reference to locality considerations, but as it requirestwo particles, such considerations might creep in. Simon et al. (2000), have mapped the Cabello/Garca-Alcaine scheme onto a combination of position and spin observables for a single particle. Their experiment has been carried out (Huang et al. 2003) and has confirmed the QM predictions. All the mentioned authors conside

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r their experimental proposals as empirical refutations of NC, but this has beendoubted (Barrett and Kent 2004), for reasons considered in the next paragraph.

(3) The KS theorem, by its mathematical nature, is not empirically testable. However, we could, along the lines of the previous paragraphs, try to measure a subset of a suitable KS-uncolourable set. Especially, it should be possible to produce cases along the lines of Clifton's example (3.5) where QM and a noncontextual HV theory make measurably different predictions. It seems as if such cases could provide empirical tests of whether Nature is contextual (though not whether such contextuality is of the causal or ontological type). It has, however, been argued that such testing is impossible. The KS theorem, it is claimed, leaves enough loopholes for a HV theory at variance with QM, but able to reproduce the theory's empirical predictions. Pitowsky (1983, 1985) has argued that it is possible to restrict attention to a subset of directions in R3 which are colourable. His argument, however, relies on a non-standard version of probability theory thatis regarded as physically implausible. More recently, Meyer (1999) has exploited the mathematical fact that a set DM of directions in R3 approximating the KS-set arbitrarily closely, but with rational coordinates is KS-colourable. Meyer argues that real measurements have finite precision and thus can never distinguishbetween a direction in R3 and its approximation from DM. Kent (1999) has generalized the result for all Hilbert spaces, and Clifton and Kent (2000) have shownthat also a set of directions DCK such that every one direction is a member of just one orthogonal triple approximates any direction arbitrarily closely. In DCKthere are no interlocking triples, the question of contextuality does not arise

and DCK trivially is KS colourable. Clifton and Kent, in addition, have explicitly shown that DCK is large enough to allow probability distributions over valueassignments arbitrarily close to all QM distributions. Meyer, Kent and Clifton(MKC) can be understood as thus arguing that even an empirical test of KS-uncolourable directions confirming the QM predictions cannot prove the contextuality of Nature. Because of the test's finite precision it is impossible to disprove the contention that unwittingly we have tested close-by members of a KS-colourableset. One quite obvious objection to this type of argument is that the originalKS argument works for possessed values, not measured values, so the MKC argument, dealing on finite precision of measurement, misses the mark. We might not be able to test observables which are exactly orthogonal or exactly alike in different tests, but it would be a somewhat strange HV interpretation that asserts thatsuch components do not exist (see Cabello 1999 in Other Internet Resources). Of

course, such a noncontextual HV proposal would be immune to the KS argument, but it would be forced to either assume that not for every one of the continuouslymany directions in physical space there is an observable, or else that there are not continuously many directions in physical space. Neither assumption seems very attractive.

In addition, the MKC argument is dissatisfying even for measured values, since it exploits the finite precision of real measurements only in one of the above senses, but presupposes infinite precision in the other. MKC assume, for measuredobservables, that there is finite precision in the choice of different orthogonal triples, such that we cannot, in general, have exactly the same observable twice, as a member of two different triples. However, MKC still assume infinite precision, i.e. exact orthogonality, within the triple (otherwise the colouring con

straints could find no application, at all). It has been claimed that this feature can be exploited to rebut the argument and to re-install contextualism (see Mermin 1999 and Appleby 2000, both in Other Internet Resources).

Finally, it seems plausible to assume that probabilities vary continuously as wechange directions in R3, so small imperfections of selection of observables that block the argument (but only for measured values!) in the single case will wash out in the long run (see Mermin 1999, in Other Internet Resources). This in itself does not constitute an argument, since in the colourable sets of observables in MKC's constructions probabilities also vary (in a sense) continuously.[17]

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We might, however, exploit Mermin's reasoning in the following way. Reconsider Clifton's set of eight directions (in Figure 3) leading to a colouring constraintfor the outermost points which statistically contradicts the QM statistics by afraction of 1/17. Using Clifton and Kent's colourable set of directions DCK weare unable to derive the constraint for the eight points, since these eight points do not lie in DCK; namely, as we move, in the colourable subset, from one mutually orthogonal triple of rays to the next, we never hit upon exactly the sameray again, but only on one approximating it arbitrarily closely. Assume a set Sof systems wherein observables, corresponding to members of DCK and approximating the eight directions in Fig. 3 arbitrarily closely, all have values in accordance with the HV premise. Then we can derive Clifton's constraint for the outermost points in the following sense. Consider the subset S S of systems where any direction approximating point (1, 1, 1) gets value 1 (or colour white). In order to meet the predictions of QM, in S all directions approximating (1, 0, 1) and (1, 1, 0) must receive values such that the probability of value 0 (or colour black)is extremely close to 1. Analogously, in another subset S S of systems with directions approximating (1, 1, 1) as having value 1 (colour white) all directions approximating (1, 0, 1) and (1, 1, 0) must receive values such that the probabilityof value 0 (colour black) is extremely close to 1. Consider now members of S S. In any of them there will be, for any approximation to (1, 0, 1) with value 0 (colour black), an exactly orthogonal point that approximates (1, 0, 1) and also hasvalue 0 (colour black) such that there is a third orthogonal point approximating (0, 1, 0) and having value 1 (colour white). Likewise for (0, 0, 1). But (0, 1, 0) and (0, 0, 1) are orthogonal, and for all members of S S the directions appro

ximating them both have value 1 (colour white), while QM predicts that the probability for values 1 for the approximated directions values is 0. In order to ensure that this prediction is met, S S must be an extremely small subset of S, whichis to say that the probability for both (1, 1, 1) and (1, 1, 1) (the leftmost and rightmost points in fig. 3) must be close to 0 and approximate 0 better and better as S grows. QM, on the contrary, predicts a probability of 1/17. (Recall also that this number can be pushed up to 1/3 by choosing a set of 13 directions!)

Cabello (2002), using a very similar reasoning, has shown that the MKC models lead to predictions that testably differ from the ones of QM. For DCK, he effectively uses the strategy sketched above: QM gives probabilities for directions in the Clifton-Kent set which their model must match in order to reproduce QM predictions. As these directions are arbitrarily close to directions from a KS-uncolou

rable set (or directions leading to Clifton's constraint), this leads to restrictions for these nearby points that are measurably violated by the QM predictions. For Meyer's DM Cabello's case is even stronger. He explicitly presents a set of nine rational vectors leading to predictions different from QM (for three of these directions). Hence, the Meyer argument is effectively rebutted (without recourse to Mermin's requirement): Even if there were only observables corresponding to the rational directions in R3 (which in itself is an implausible assumption) a theory assuming that they all have noncontextual values faithfully revealedby measurement will be measurably at variance with QM. Assume now that the Cabello directions were tested and the QM predictions reliably confirmed, then this would (modulo the reliability of the tests) constitute a proof that Nature is contextual.

So, in sum it seems that, as long as we assume that there are continuously manyQM observables (corresponding to the continuum of directions in physical space),statistical tests building, e.g., on the Clifton 1993 or the Cabello/Garca-Alcaine 1998 proposal remain entirely valid as empirical confirmations of QM and, viathe KS theorem, of contextuality. Since these statistical violations of the HVprogramme come about as contradictions of results of QM, VD, VR, and NC on the one hand, and QM and experiment on the other, the experimental data still force upon us the trilemma of giving up either VD or VR or NC. As we have seen, denialof value realism in the end becomes identical to a kind of contextualism, hencewe really have only two options: (1) Giving up VD, either for all observables fo

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rbidden to have values in the orthodox interpretation (thus giving up the HV programme, as defined above), or for a subset of these observables (as modal interpretations do). (2) Endorse a kind of contextualism. Moreover, as things presently stand, the choice between these two options seems not to be a matter of empirical testing, but one of pure philosophical argument.