Let me start this section with saying that the MKC models are highly artificial. As a candidate for an interpretation of quantum mechanics or a route to ‘new physics’, they perform a terrible job. Their value lies purely in their capacity to provide insight in the notion of contextuality in quantum mechanics, and the relation between quantum and classical probability. It is not my intention here to perform a re-hash of the pros and cons of these models, nor on the precise metaphysical meaning of contextuality in quantum mechanics. For this I refer the reader to the discussions in (Barrett and Kent, 2004; Appleby, 2005; Hermens, 2011) and references therein. Instead, I focus on some particular aspects of the way they perform as classical representations of quantum probability.
The finite precision argument may be seen to rest on two ingredients. The first is the uncontroversial idea that it is impossible to determine with infinite precision the parameters that fix the experimental setup associated with an observable. In- deed, if this were possible, the MKC models would not be empirically equivalent to quantum mechanics. One could ‘just’ check for any particular direction r if there is an observable corresponding to S2r. The scare quotes around ‘just’ indicate how implausible such a test is, making the finite precision assumption a very plausible one.
The second ingredient relates to the central theme of this section: continuity. To make the finite precision argument precise one needs a metric (or at least a topology) to specify when two experimental setups resemble each other. The assumption CoO states that this should be done by adopting the metric on the set of self-adjoint operators. This metric is plausible for spin-measurements: the two operators Sr21 and Sr21 are close to each other just in case the axes spanned byr1 andr2 are near
each other. But in other situations its relevance may be less clear. For example, the metric also provides a distance between position and momentum measurements, and it is not clear how this relates to the experimental setups. However, this need not pose a difficulty. For CoO the distance between the position and momentum operator is only relevant when adopting the further assumption that there is a whole range of observables in between. It is this further assumption that is troublesome, rather than CoO itself. In fact, CoO is a relatively innocent assumption because it is formulated as a conditional.
But also when the finite precision assumption and CoO are accepted, there are reasons to reject the MKC models as faithful classical representations. I discuss here two reasons. The first is the by now familiar complaint that not all probability functions on Λ(P)i are of the form Pρ. However, in this case there is a formal
argument available that selects the quantum probability functions as a natural subset of all probability functions. After giving this argument below I discuss the second reason, which explains why CoO is an unsatisfactory assumption from the point of view of the MKC models themselves. Thus the MKC models undermine the very assumption that allows their construction.
7.5.1 Deriving the Born rule
The introduction of CoO was motivated by the idea that a similar claim tends to hold in quantum mechanics itself. This was illustrated by Theorem 7.2. The explanation of the special role of quantum probability functions in the MKC models given in this section uses the converse fact that, if probability functions satisfy a certain continuity assumption, they satisfy the Born rule. Thus, roughly speaking, probability functions in the MKC models satisfy the Born rule if and only if they are continuous. This statement will be made more precise below, after giving a possible defense for singling out continuous probability functions.
A particular way to motivate a continuity assumption for probability functions is by adopting an epistemic interpretation of probability.8 That is not to say that such an interpretation is necessary here, but at least it provides a useful way to think about the formal side of the story. On an epistemic reading, probabilities are thought of as epistemic judgments of a rational agent concerning outcomes of particular measurements. However, when considering the outcomes of a particular actual measurement the agent not only has to reflect on the uncertainty pertaining to the outcome of the measurement, but also (due to the finite precision argument) on the uncertainty pertaining to which measurement is actually being performed. For example, consider the measurement of an observable corresponding toSr2. The agent is required to assign probabilities to the possible outcomes 0 and 1 for every possible value ofr. But apart from the uncertainty about the outcome, the agent is also necessarily uncertain about the value ofr. In order for this notion of probability to be made operational, the probabilities assigned to the possible outcomes 0 and 1 have to vary continuously with r. For example, when thinking of probabilities as betting ratios, one has to be able to agree on what the payoff will be after a measurement of Sr2 without requiring that one determines with infinite precision
8Inspiration for this kind of motivation for a continuity assumption is drawn from the work of
The role of continuity
7.5
what the value ofr is. Thus one has that
lim r0→rP FS2 r0 = 1 =P FS2 r = 1 . (7.33)
When taking into account CoO, this leads to the demand thatPshould be contin- uous with respect to the metric on the self-adjoint operators. Here I propose the following definition to make this precise.
Definition 7.2. A probability measure P on ΛMKC is said to respect CoO, if for
everyA ∈ OMKC and every >0 there exists a δ > 0 such that for allA0 ∈ OMKC
withσ(A0) =σ(A),
kA−A0k< δ ⇒ |P(FA=a)−P(FA0 =a)|< (7.34)
for all a∈σ(A).
One can now show that the Born rule precisely singles out these continuous probability distributions. Conversely, from the perspective of the MKC models, the Born rule can be seen as a consequence of the continuity assumption. Note that al- though this continuity assumption may be justified with an epistemic interpretation of probability, the formal result is independent of the interpretation of probability.
Theorem 7.4. A probability measureP onΛM KC respects CoO if and only if there
is a density operator ρ such that
P(FA∈∆) = Tr (ρµA(∆)) (7.35)
for all A∈ OM KC and ∆⊂σ(A).
Proof. Note that ifPis given by a density operatorρ, then it follows from Theorem 7.2 that CoO is respected. It thus only has to be shown that, if P respects CoO, then there exists a density operatorρsuch that (7.35) holds. LetPbe a probability function that respects CoO. This function can be used to define a function λP : L(H)→[0,1] in the following way. LetP ∈L(H). According to Theorem 7.3 there is a sequence (Pi) in L(H)∩ OMKC such that Pi →P asi→ ∞. Now define
λP(P) := lim
i→∞P(FPi = 1). (7.36)
Because P respects CoO this value is independent of the choice of the sequence. Thus the function λP is well-defined. Further, again because of CoO, this function satisfies λP Pi∈NPi
= P
i∈NλP(Pi) for every countable sequence of pairwise or-
thogonal projection operators. Since it is also the case thatλP(1) = 1 this defines a quantum probability function (see Definition 5.1). Finally, it follows from Gleason’s theorem that (7.35) holds.
It deserves to be emphasized that Theorem 7.4 is quite a remarkable result. It establishes success on a point where previous classical representations have failed: to derive the Born rule from within the classical model. Moreover, it strengthens a recurring idea in results on reconstructing quantum mechanics such as (Hardy, 2001), namely, that continuity is an essential ingredient. But the result also puts additional weight on the role of the assumption CoO, and it deserves to be investi- gated how well this assumption fits with the construction of the MKC models. The conclusion will be that it isn’t a very attractive assumption.
7.5.2 The incompatibility with continuity
CoO establishes that the metric onOsa also plays a role on the set of observables
OMKC of the MKC models. This adds additional structure of the Hilbert space
formalism to the MKC models, on top of the functional relationships EFR. But from the point of view of the MKC models, this additional structure on the set is somewhat alienating. In the construction of Λ(Pi) all relations between frames
are placed on an equal footing. A state λ is blind to the distance between two observables whenever their corresponding operators do not commute. To put it another way, the MKC states do naturally reproduce EFR, but there is nothing in the MKC models that could even lead to the formulation of CoO.
The tension would be relieved if the metric could in some way be reflected within the properties of the MKC states. This would be the case if, typically, for observables that are close to each other, the values assigned to them are close to each other as well. The provisional ‘typically’ is to indicate that a too stringent demand is doomed to fail from the start. For example, in the 3-dimensional case, when restricting an MKC stateλto the squared spin observablesSr2, this function cannot be a continuous function of r. This is because the range of the function is the discrete set {0,1}, and both values have to be attained. Some discontinuities are therefore necessary. But there is still the possibility that these can be ‘special’ and that ‘generally’ the value ofλ(Sr2) is a good indication of the values of λ(Sr20)
forr0 close tor. It was shown by Appleby (2005) however, that this is not possible: there have to be wild discontinuities for every MKC state.
The theorem proven by Appleby is formulated in terms of coloring functions. Let S2 denote the 2-sphere whose points represent the squared spin observables.
Now set
S2MKC :=
r ∈S2Sr2 ∈ OMKC . (7.37)
This is a dense colorable subset ofS2. In fact, every λ∈ΛMKC provides a coloring
functioncλ via
The role of continuity
7.5
Theorem 7.5 (Appleby 2005). Let OMKC be an MKC set of observables for the Hilbert space C3. Then for every λ∈ΛMKC there exists an open set U ⊂ S2 such that cλ is discontinuous at every point ofU ∩S2MKC.
Although the theorem pertains to spin-1 systems specifically, it is likely to be generalizable to higher dimensional systems. For Appleby, the importance of this result lies in the implication that, in general, measuring some observable Adoesn’t provide any information about the state λof the system. It doesn’t do so in prin- ciple, whenever the measurement is of an observable A in a region in which λ is densely discontinuous. But also doesn’t do so pragmatically, since there is no way to have empirical access to the question whether A is in such a region or not. The broader lesson Appleby draws from the (Bell-)Kochen-Specker theorem then is not so much on the (im)possibility of certain kinds of hidden variables. Rather, the theorem poses a limit on the extent to which such hidden variables are empirically accessible. Thus Appleby reads in the Kochen-Specker theorem an early version of what more recently Colbeck and Renner (2012b) aimed to show: that quantum mechanics is ‘maximally informative’. Here though I am more interested in the implications of Theorem 7.5 for the MKC models and the classical representations based on them.
As mentioned at the beginning of this section, the MKC models are unsatis- factory candidates for a physical theory. One peculiarity about them is that the observables do not form a continuum. This is a big break with scientific theories of the past and present. In particular, there is a tension with the common conception of space being adequately modeled by R3. This in itself need not be a reason to
be discontent with the models. A departure from this conception already occurs of course in general relativity, and in quantum gravity one may even move to a discrete conception of space. But it is problematic that the departure is ill-motivated. The set of operators OMKC used in section 7.4 is selected in an ad hoc way, and there
is no unique way to construct it. Consequently, the departure from a continuum is not as simple and clear as going from R3 to Q3.9 But what is most disturbing is
that Theorem 7.5 shows that this departure causes a particular tension within the models themselves. The finite precision argument requires that observables appear to form a continuum by CoO. But this apparent continuity is nowhere coded into the theory: it is neither there at the level of the random variablesOMKC, nor at the
level of the MKC states Λ(Pi). Theorem 7.5 shows this is necessarily so.
This tension in the MKC models (i.e., requiring a continuity assumption while at the same time not being able to respect this assumption) is not just a moot point. The assumption CoO is not only used in the finite precision argument, but also in
9
This is in contrast with the paper of Meyer (1999) in which spin-directions are associated with directions inQ3. Strictly speaking though, this does not give an MKC model, as the hidden
the derivation of the Born rule in Theorem 7.4. While this result is interesting, the fact that Λ(Pi)does not respect CoO indicates that the theorem is not as important
as it pretends. Difficulties arise, in particular, when one considers update rules for probability functions in the MKC models. Suppose one measures A and finds the result a. The usual way to update the probability function is by conditioning on the set of all λ with FA(λ) = a. Now, because FA is independent of FA0 for
every A0 with [A, A0] 6= 0, this update has a negligible impact on the probability distribution. It only changes the probability distribution for one of the countably many frames that make up ΛMKC (see (7.32)). This may be seen as a vindication
of Appleby’s objection that one can’t learn anything about the hidden variables from a measurement. On top of that, one may note that the probability function one obtains by conditionalization is radically different from what one would get by using the projection postulate.
There are also some other problems with conditionalization as an update rule. For one, even if one started with a probability function that respects CoO (and thus is quantum like), after conditionalization it no longer respects CoO. Namely, for all the A0 with [A, A0] 6= 0, the updated probability function still satisfies the trace rule, while forA the probability is now peaked around the value a. Another problem is that, due to the finite precision argument, one cannot know exactly for which A to perform the update. A naive solution could be to say that, since one knows roughly which A has been measured, and one knows that a repeated measurement ofA would yield the same result (assuming λ wasn’t influenced too much by the measurement), one should assign high probability to the set of allλwith FA0(λ) =afor all A0 close to A. Effectively, this would mean applying (something
very much resembling) the projection postulate. But although one could hope that this process saves the Born rule, the proposed update is completely unwarranted from the perspective of the MKC models. In general, the value of λ(A0) need not resemble λ(A), and the existence of densely discontinuous regions indicates that changing the values ofλ(A0) will often lead to false predictions.
In brief, the statistical predictions for the MKC models for consecutive mea- surements are problematic to define and possibly contradict quantum mechanical predictions. This is not to say it is impossible to define such measurements without contradictions. In (Hermens, 2011) I proposed the following artificial solution to the problem. Take the MKC states to specify the state of the system at any point in time, but let these states evolve stochastically in time. Specifically, at any in- stant the MKC state is selected in accordance with the Born rule, and the quantum state guiding this stochastic process evolves according to the dynamics of quantum mechanics. On this approach, the Born rule is applied as a postulate, rather than being derived from the assumption CoO. In essence, this MKC model is just or- thodox quantum mechanics with some decorative fluff that is neither attractive nor explanatory. This works fine for showing the logical possibility of non-contextual
Taking stock
7.6
hidden variables. However, as a classical representation of quantum probability, the models recover the old difficulty of being incapable of explaining the special role of the probability functions that satisfy the Born rule.