The Bell–Kochen–Specker theorem
Introduction
In ordinary language the word “measurement”
The Bell–Kochen–Specker (Bell-KS) theorem (Bell, 1966, Kochen and Specker, 1967) shows that quantum mechanics is inconsistent with that natural idea.very strongly suggests the ascertaining of some pre-existing property of some thing, any instrument involved playing a purely passive role (Bell, 1987, p. 166).
Or so Bell thought. His conclusion has, however, been challenged: first by Pitowsky, 1983, Pitowsky, 1985, and then, using a different set-theoretic argument, by Meyer (1999), Kent (1999) and Clifton and Kent (2000), Meyer, Kent, and Clifton (MKC in the sequel). MKC's argument was inspired by the previous work of Hales and Straus (1982) and Godsil and Zaks (1988). It has attracted much comment (see Mermin, 1999, Havlicek et al., 2001, Simon et al., 2001, Larsson, 2002, Appleby, 2001, Appleby, 2002, Cabello, 2002; Breuer, 2002a, Breuer, 2002b).
MKC claim to have “nullified” the Bell-KS theorem. They mean by this that the theorem, though mathematically valid, is not physically significant. Pitowsky expresses himself less forcefully. He does not say, in so many words, that the Bell-KS theorem is entirely without significance. However, he clearly means to insinuate doubts.
Now there can be no question as to the importance of the results proved by Pitowsky and MKC (PMKC in the sequel). They clearly have some major consequences. However, we will argue that these consequences are less catastrophic than MKC think. They do not show that the Bell-KS theorem is without significance. They only show that we need to reassess its significance.
The question is complicated by the fact that the Bell-KS theorem was proved twice over: first by Bell (1966), and then again by Kochen and Specker (1967). They expressed strikingly different views as to the theorem's physical significance. So the first thing one needs to ask is: what, exactly, is it that the MKC models are supposed to nullify?
In Section 8 we will argue that PMKC have indeed invalidated Kochen and Specker's account of the theorem's significance. We will also argue that Bell's account contains a number of serious misconceptions. So it is true that PMKC nullify some of what was previously seen as the theorem's significance. But they do not nullify it all. In particular, they do not nullify Bell's main point, as stated above. It remains the case that quantum mechanics is inconsistent with classical ideas about measurement.
The fact that PMKC cannot have fully restored the ordinary, or classical concept of measurement becomes obvious, as soon as one reflects that their models are still hidden variables theories. A hidden variables theory is one in which the pre-existing values are, for some reason, concealed. But if the observables could be measured in the ordinary, classical sense, then the values would not be concealed. They would be open-to-view, as in classical physics. It follows that, in a hidden variables theory, there must necessarily be some breakdown of classical assumptions about measurement.
In Bohmian mechanics the mechanism responsible for concealing the values is the same as the mechanism which makes the theory contextual. Instead of the instrument playing a purely passive role, as it would classically, there is a complex interplay between system and instrument. The effect is to create a value, which did not exist before. As Bell puts it:
(for a more detailed discussion of this point see Dewdney et al., 1986, and Holland, 1993).The result of a ‘spin measurement’, for example, depends in a very complicated way on the initial position of the particle and on the strength and geometry of the magnetic field. Thus, the result of the measurement does not actually tell us about some property previously possessed by the system, but about something which has come into being in the combination of system and apparatus. (Bell, 1987, p. 35)
PMKC have discovered a completely different mechanism for concealing the values. This is an important discovery. It means, in particular, that Bell's emphasis on the active role of the apparatus needs revision. Nevertheless, their models still are hidden variables theories. So Bell's main point, that a measurement outcome “does not actually tell us about some property previously possessed by the system,” still stands.
The variables are hidden in the PMKC models because their colourings are violently, and even “pathologically” discontinuous (the relevance of continuity in this regard is noted by Mermin, 1999). Consider, for instance, the colouring described by Meyer (1999). This function is discontinuous at every point in its domain of definition. It is, in other words, as discontinuous as a function can possibly get.
In a conventional colouring, such as a political map of the Earth, the paints are applied in broad strokes to well-behaved regions having regular boundaries. This means that it is almost always possible, using finite precision measurements, to find out what country one is in. It is true that infinite precision would be needed if one was situated exactly on a boundary. However, there is zero probability of hitting such a point.
Meyer's colouring is not like that. In the Meyer colouring it is as if, before applying the paints, one first mixes them, to the maximum extent possible, so that the colours become intermingled at the molecular level. It is (so to speak) a maximum entropy colouring. This makes the colours unobservable.
At least, the colours are not observable using finite precision measurements. MKC say that finite precision nullifies the KS theorem. But it would be nearer the truth if one said that finite precision saves the KS theorem.
Suppose one tries to find out the value of some particular vector (where is the rational unit 2-sphere, on which the Meyer colouring is defined) using a finite precision measurement. The finite precision means that the measurement actually reveals the value of some unknown vector , with for some positive . If the colouring were continuous at , and if were sufficiently small, then the value of would be the same as the value of . However, the colouring is, in fact, discontinuous. This means that, no matter how small the error , the measurement provides no more information about the value of than could be obtained by simply guessing a number at random (assuming that is not actually 0). The value is therefore unobservable, or hidden.
A procedure which leaves the experimenter in complete ignorance of the pre-existing values is clearly not a measurement in the classical sense. MKC focus on the point that, in their models, a measurement does always reveal the pre-existing value of something: namely, the value of the vector representing the true alignment of the instrument. What they overlook is that the experimenter does not know the true alignment of the instrument. This means that, although the experimenter learns a value, s/he has no idea what it is a value of. Consequently, the experimenter does not acquire any actual knowledge.
A classical measurement is not simply a procedure which reveals a pre-existing value. Rather, it is a procedure which ascertains a pre-existing fact, of the form “observable A had value x”. The specification of the observable A is no less essential than the specification of the value x. The problem with the PMKC models is that the observable A is not specified, so the experimenter only learns “ had value x”. This statement is completely uninformative. It says no more than the statement “observable A had ”. Indeed, it says no more than the completely empty statement “ had ”.
What emerges from this is that PMKC have been asking the wrong question. The important question is not: “How much of (the full unit 2-sphere) can be coloured at all?” But rather: “How much of can be coloured in such a way that the colours are empirically knowable?”
In Sections 2–7 we address that question. We have seen that, in the case of Meyer's model, none of the colours are empirically knowable. We need to consider whether another model might improve on that.
So as to have a standard of comparison we begin, in Section 2, by defining the concept of a regular KS-colouring. Intuitively, this is a colouring where the paint is applied in broad strokes, as in a political map of the Earth. We show that a regular KS-colouring must exclude a region having non-empty interior and subtending solid angle . Here is a fixed positive number whose value is determined, once and for all, by the principles of quantum mechanics.
We refer to functions of the same general kind as the PMKC colourings as pseudo-KS-colourings of . In Section 3 we identify two conditions satisfied by every PMKC colouring (both the ones constructed by Pitowsky and the ones constructed by MKC). We argue that they would also have to be satisfied by any other pseudo-KS-colouring.
In Sections 4–5 we use these conditions to analyze the discontinuities of an arbitrary pseudo-KS-colouring. We show that splits into an open set U, which is regularly KS-colourable, and a closed set D on which the discontinuities make the colours empirically unknowable. In the case of Meyer's colouring U is empty and . In the general case D might be smaller. However, it cannot be shrunk to nothing. The fact that U is regularly KS-colourable means that D must always have non-empty interior, and subtend solid angle .
In Section 6 we infer that the Bell-KS theorem is not nullified. A conventional colouring must exclude a region D which is simply not coloured at all. PMKC have found ways to extend the colouring into D. However, they only do so at the price of making the valuation so extremely discontinuous that the colours are empirically unknowable. From the point of view of a finite precision experimenter, who wants to ascertain the pre-existing values, this is not an improvement.
The fact that the colours cannot all be empirically knowable is also shown by Cabello (2002). However, the relation between Cabello's argument and ours may not be immediately apparent. In Section 7 we elucidate the relationship.
Finally, in Section 8 we assess the implications of PMKCs discoveries, and our counter-argument. PMKC clearly nullify some of what used to be seen as the Bell-KS theorem's significance. We will argue that they completely invalidate what KS say on the subject. They also invalidate some of the things said by Bell. In addition we present some further criticisms of Bell, which are only indirectly inspired by PMKC's argument. In short, PMKC make us recognize that we did not, in the past, fully understand what the theorem is telling us.
However, none of this detracts from the theorem's importance. The failure of classical assumptions about measurement is arguably the single most revolutionary feature of quantum mechanics.
Section snippets
Regular KS-colourings
We begin by showing that there is certainly no question of nullifying the Bell-KS theorem by means of the kind of well-behaved colouring one sees in a political map of the Earth. The results proved in this section will also play an important role in our subsequent analysis of the PMKC models.
The Bell-KS theorem states that there is no valuation (where is the unit 2-sphere) such that for all andfor every triad (every triplet of orthogonal unit
Pseudo-KS-colourings
We refer to functions like the ones constructed by PMKC as pseudo-KS-colourings of . In this section we identify two conditions which the PMKC colourings all satisfy (both the ones constructed by Pitowsky and the ones constructed by MKC). We argue that they would also have to be satisfied by any other pseudo-KS-colouring.
We aim to give a completely general analysis, applying to any pseudo-KS-colouring. But let us begin by looking at the particular colourings constructed by PMKC.
The clearest
The phenomenological colouring
The mathematical properties of a pseudo-KS-colouring are best visualized using a three-coloured variant of the usual chromatic metaphor. We will call this the phenomenological colouring. Intuitively, it describes what is seen when the sphere is viewed through finite resolution eyes.
Consider some fixed pseudo-KS-colouring . Let be the set of points f-evaluating to 0, and let be the set of points f-evaluating to 1.
We begin by defining a true, or intrinsic
The discontinuity region
We now use the phenomenological colouring to investigate the discontinuities of f.
Let us begin by defining the phenomenological colouring in more formal terms. Let (respectively ) be the closure of (respectively ) considered as a subset of . Then . Now define , and . Then , , D partition into three pairwise disjoint subsets. Furthermore, , are open and D is closed.
The phenomenological colouring may now be defined
The Bell-KS theorem is not nullified
In a regular KS-colouring there are some remaining patches of white, which are simply not coloured at all. A pseudo-KS-colouring replaces these patches of white with patches of black, on which the colours are defined, but empirically unknowable. From the point of view of a finite precision experimenter, who wants to ascertain the intrinsic colour of a specified point, this is not an improvement.
Conventional models, such as the Bohm theory, and unconventional models, such as the ones proposed by
Cabello's argument
Cabello (2002), in an important paper, has given an argument which is closely related to ours. Suppose an experimenter makes a finite precision measurement in the direction with alignment uncertainty . Let be the probability that the measurement reveals the true colour of . Classically, one would assumeCabello, however, shows5
The physical significance of the Bell-KS theorem
The PMKC models show that the significance of the Bell-KS theorem is primarily epistemological: it concerns the nature and extent of the knowledge acquired by measurement. In this section we examine how far that proposition departs from the views of Bell and KS. We also explain why, in our view, the theorem matters: why it deserves its status as one of the key foundational results of quantum mechanics.
For a long time it was widely (though not universally) believed that quantum mechanics is just
Conclusion
PMKC have made a most important contribution to this subject. However, it is not important for the reason MKC think. The PMKC models do not nullify the Bell-KS theorem. Instead, they give us a deeper and more accurate insight into what the theorem is really telling us.
We have argued that the Bell-KS theorem has a primarily epistemological significance. It concerns the knowledge we acquire by measurement. So what one needs to ask is not: “how much of can be coloured at all?” But rather: “how
Acknowledgements
The author is grateful to H. Brown, J. Butterfield, A. Cabello, R. Clifton, C. A. Fuchs, A. Kent, J.-Å. Larsson, N. D. Mermin, T. N. Palmer, A. Peres, I. Pitowsky and K. Svozil for useful discussions. He is also grateful to two anonymous referees, for some very helpful comments.
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