The Bell–Kochen–Specker theorem

Abstract

Meyer, Kent and Clifton (MKC) claim to have nullified the Bell–Kochen–Specker (Bell-KS) theorem. It is true that they invalidate Kochen and Specker's account of the theorem's physical implications. However, they do not invalidate Bell's point, that quantum mechanics is inconsistent with the classical assumption, that a measurement tells us about a property previously possessed by the system. This failure of classical ideas about measurement is, perhaps, the single most important implication of quantum mechanics. In a conventional colouring there are some remaining patches of white. MKC fill in these patches, but only at the price of introducing patches where the colouring becomes “pathologically” discontinuous. The discontinuities mean that the colours in these patches are empirically unknowable. We prove a general theorem which shows that their extent is at least as great as the patches of white in a conventional approach. The theorem applies, not only to the MKC colourings, but also to any other such attempt to circumvent the Bell-KS theorem (Pitowsky's colourings, for example). We go on to discuss the implications. MKC do not nullify the Bell-KS theorem. They do, however, show that we did not, hitherto, properly understand the theorem. For that reason their results (and Pitowsky's earlier results) are of major importance.

Introduction

In ordinary language the word “measurement”

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).

The Bell–Kochen–Specker (Bell-KS) theorem (Bell, 1966, Kochen and Specker, 1967) shows that quantum mechanics is inconsistent with that natural idea.

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:

The result of a ‘spin measurement’, for example, depends in a very complicated way on the initial position $\lambda$ 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)

(for a more detailed discussion of this point see Dewdney et al., 1986, and Holland, 1993).

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 $\mathbf{n}\in S_{\mathbb{Q}}^2$ (where $S_{\mathbb{Q}}^2$ 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 ${\mathbf{n}}^{\prime }$, with $|{\mathbf{n}}^{\prime }-\mathbf{n}|⩽\epsilon$ for some positive $\epsilon$. If the colouring were continuous at $\mathbf{n}$, and if $\epsilon$ were sufficiently small, then the value of ${\mathbf{n}}^{\prime }$ would be the same as the value of $\mathbf{n}$. However, the colouring is, in fact, discontinuous. This means that, no matter how small the error $\epsilon$, the measurement provides no more information about the value of $\mathbf{n}$ than could be obtained by simply guessing a number at random (assuming that $\epsilon$ 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 ${\mathbf{n}}^{\prime }$ 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 “ $\dots$ had value x”. This statement is completely uninformative. It says no more than the statement “observable A had $\dots$”. Indeed, it says no more than the completely empty statement “ $\dots$ had $\dots$”.

What emerges from this is that PMKC have been asking the wrong question. The important question is not: “How much of $S^2$ (the full unit 2-sphere) can be coloured at all?” But rather: “How much of $S^2$ 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 $⩾4\pi d_{\mathcal{R}}$. Here $d_{\mathcal{R}}$ 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 $S^2$. 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 $S^2$ 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 $D=S^2$. 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 $⩾4\pi d_{\mathcal{R}}$.

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.