The Bell–Kochen–Specker theorem

Abstract

Meyer, Kent and Clifton (MKC) claim to have nullified the Bell-KochenSpecker (Bell-KS) theorem. It is true that they invalidate KS'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.

Non-contextuality, finite precision measurement and the Kochen–Specker theorem

“…As Appleby (Appleby, 2001(Appleby, , 2003 and Ca-bello (Cabello, 2002) show, they are. 7 Appleby's argument thus cannot be sidestepped.…”

“…The arguments set out by Meyer, Kent, and Clifton and Kent (MKC) have evoked some controversy (see, e.g., Mermin, 1999;Cabello, 1999;Basu et al, 2001;Simon et al, 2001;Larsson, 2002;Appleby, 2000Appleby, , 2001Appleby, , 2002Appleby, , 2003Havlicek et al, 2001;Cabello, 2002;Breuer, 2002) and even a parodyPeres (2003). Among other things, it has been suggested (Cabello, 2002) that the CK models do not in fact reproduce correctly the predictions of quantum mechanics, even when finite precision is taken into account.…”

“…Some (Appleby, 2000(Appleby, , 2001(Appleby, , 2003Havlicek et al, 2001) have queried whether the models can, in fact, properly be described as classical, given that they define values on dense subsets of the set of measurements in such a way that every neighbourhood contains operators with both truth values. This feature implies that the models do not satisfy what we call the faithful measurement condition: that one can in general ascribe a value to an operator P , such that this value, or one close to it, is obtained with high probability when a high precision measurement of P is performed.…”

“…Appleby (see Appleby, 2000Appleby, , 2001Appleby, , 2003 has discussed the faithful measurement condition at some length, arguing that it is a necessary property of measurements in classical models. Appleby notes that a classical measurement tells us some definite fact about the system as it was before measurement, and goes on to argue that the dense -in Appleby's words, "radical" or "pathological" (Appleby, 2001(Appleby, , 2003 -discontinuities of truth values in the CK models mean that they cannot satisfy this epistemological criterion: let us call it the definite revelation criterion.…”

“…Appleby notes that a classical measurement tells us some definite fact about the system as it was before measurement, and goes on to argue that the dense -in Appleby's words, "radical" or "pathological" (Appleby, 2001(Appleby, , 2003 -discontinuities of truth values in the CK models mean that they cannot satisfy this epistemological criterion: let us call it the definite revelation criterion.…”