The MKC models are a specific kind of attempt to demonstrate the logical possibility of non-contextual hidden variable theories. Noticeable other attempts are the work of Pitowsky (1983 ; 1985) and La Cour (2009). The first one will not be discussed here because it adopts a peculiar kind of non-classical probability in which, for example, the conjunction of two probability-1 events can have probability zero. Such a model can hardly be considered a classical representation. The second manages to uphold both NC and IP by rejecting FM. Violations with quantum mechanical predictions (such as EFR) are circumvented by adopting a peculiar dynamics. This departure from static considerations takes us too far adrift from the discussion of classical representations and the theory will therefore not be discussed.
Before going into the finite precision argument, it is good to briefly recapitulate what is formally at stake. The assumptions NC and IP both concern the relation between operators and observables. Both presuppose the validity of the observ- able postulate (OP) of quantum mechanics, which states that observables can be associated with operators. Specifically, it postulates the existence of a function f :Obs → Osa. This function establishes that formal proofs concerning operators
can have conceptual implications. However, the postulate does not specify what the function is, and this provides a gap for evaluating the philosophical implications of the formal proofs.
Formal proofs have conceptual implications only to the extent that they also apply to the relevant subset f(Obs) ⊂ Osa. The difficulty of this construction is that there is no definite way to determine precisely what this subset is and thus one requires further assumptions. Kochen and Specker showed that there is a specific finite setOKS ⊂ Osa such that if the functionf satisfies
OKS⊂f(Obs)⊂ Osa, (7.6)
then at least one of the assumptions QMKS, HV, or NC has to fail. Many results
have been presented since the original proof that aim to make OKS as small as
possible.2 What is left open in this competition is the converse question. How big
2
can a set of operatorsOMKC be such that if f satisfies
f(Obs)⊂ OMKC ⊂ Osa, (7.7)
then QMKS, HV and NC can be satisfied? This is the question Meyer (1999), Kent
(1999) and Clifton and Kent (2001) (MKC) focused on, and it turns out that, at least for finite-dimensional Hilbert spaces, this set can be surprisingly big.3
From a purely logical perspective one seems to arrive at a stalemate here. If f satisfies OKS ⊂ f(Obs), then non-contextual hidden variables are not possible, as shown by the Kochen-Specker theorem. But if on the other hand f satisfies f(Obs) ⊂ OMKC, then they are possible, as shown by MKC. The tie-breaker is provided by the finite precision argument, which favors the second option. Before explaining this argument in its general form, I first return to the spin-1 particles that figure prominently in the Kochen-Specker theorem.
It is useful to think of measurements of spin-1 particles as being analogous to measurements of spin-12 particles.4 A measurement ofσr(see Example 3.1) is usually
done with a Stern-Gerlach device which is then aligned to the axisr. The idea that every operator in the set σr
r∈R3, krk= 1 corresponds to an observable is
thus motivated by the idea that a macroscopic measurement device can be oriented in all possible ways in space. The same view may be used to motivate that all the operators in the set
Ospin:=
Sr2r ∈R3, krk= 1 (7.8)
correspond to observables. Then any finite subset, likeOKSin (7.1), does so as well. The finite precision argument counters this conclusion. To determine whether one measures the squared spin along some axis r1 or r2, one has to determine
whether some macroscopic device is aligned along either axis. But this direction can, of course, only be determined up to some finite precision. Thus for any direction rthere is always some >0 for which one cannot determine which of the directions
size also depends on the dimension of the Hilbert space under investigation. In three dimensions Conway and Kochen hold the record for the smallest number of vectors (31) (Peres, 2002, p. 114). The constructions of Peres (ibid., p. 198) and Bub (1996) both use 33 vectors of which the latter requires the least number of frames of all these results. If instead of vectors the number of frames is to be minimized, then the record is held by Lisonˇek et al. (2014) using only 7 frames for the Hilbert spaceC6 . For more discussion on comparing sizes of Kochen-Specker sets see (Bengtsson, 2012; Paviˇci´c et al., 2005) and references therein.
3
It deserves to be noted that this question was also investigated by Bub and Clifton (1996), albeit from a different perspective.
4
The reason for this little side-track is that I do not know of any actual experiments on spin- 1 particles. Actual spin-1 particles such as the W and Z bosons or Higgs bosons are hard to manipulate, and experiments on similarly behaving systems (e.g. qutrits composed of photon pairs such as in (Lanyon et al., 2008)) also have a long way to go in comparison to the manipulation of qubit systems. On the other hand, the analogy with spin-1
2 measurements may be quite apt (Swift
The finite-precision argument
7.3
r0 with kr−r0k < actually applies to the observable under investigation. The recognition of the practical underdetermination of directions was vividly exploited by Meyer (1999), who showed that it is possible to assign definite values to the operators in the set
Ospin0 :=S2rr ∈Q3, krk= 1 (7.9)
in a non-contextual way. Then, to determine whether a certain observableSr2 corre-
sponds to an operator inO0spinor in Ospin\O0spinrequires empirically distinguishing
rational numbers from irrational numbers. As this is generally considered to be im- possible, there is no empirical reason to assert that there are operators inOspin\O0
spin
that do correspond to observables. The finite precision argument then is that, be- cause one cannot determine with infinite precision which operator precisely should correspond to the observable (experimental setup) under investigation, one may as well assume that the operator is an element of a particular dense subset (O0spin) of all the operator candidates (Ospin).
The argument may come off as being contrived. The idea thatR3 is a good (ap-
proximate) model of our spatial surroundings is deeply entrenched in our scientific thinking. To replace it withQ3 then seems a desperate step just to save NC. There
are several ways to respond to this. The most neutral one is that we are interested here in logical possibilities that need not be physically attractive. But somewhat more boldly, one can argue that holding on to R3 is itself a desperate step to save
the standard reading of the Kochen-Specker theorem. And finally, one can deny the step going from the idea thatR3 correctly models space, to the conclusion that the
squared spin along the axis r is an observable for every spatial direction. Perhaps macroscopic measurement devices make an infinitesimal shift in alignment just to line up with some direction inQ3 in the case of a measurement.
The finite precision argument as it is used here works for spin observables specifi- cally. In this argument the fact is used that the observablesSr21 andS
2
r2 will resemble
each other whenr1andr2 start to approach each other. That is, whenever the oper-
atorsSr21 andSr22 are close to each other, it is used that the experimental setups used to measureSr21 andS
2
r2 are also close to each other in some sense. What is required
to make the finite precision argument work in arbitrary cases, is that something similar applies in general. That is, the metric onObs defined by5
d(A1,A2) :=kA1−A2k (7.10)
has to be relevant in the following way:
CoO (Continuity of Observables) For any observable A ∈Obs and sequence of observables A1,A2, . . ., if d(An,A) tends to zero as n → ∞, then there is a
5There is nothing stringent concerning the choice of this particular metric because the finite
sense in which the sequence of experimental setups associated with the An
will resemble an experimental setup suitable for the measurement of A as n→ ∞.
The assumption is one that I think is tacitly accepted in quantum mechanics. It is part of the application of OP to select/construct the function f : Obs → Osa
in such a way that CoO comes out true. A closely related fact is that in quantum mechanics, when one tries to pinpoint the exact operator to be identified with a particular observable, there is some wiggle room due to the finite precision with which experimental setups can be specified. This is a consequence of the fact that, for two operators A1, A2 that are close to each other, the quantum mechanical
predictions also resemble each other. For example, the following theorem holds.
Theorem 7.2. Let A be a self-adjoint operator acting on a finite-dimensional Hilbert space, and let (An)n∈N be a sequence of self-adjoint operators such that
σ(A) =σ(An) for all n and limn→∞kA−Ank= 0. Then for every ∆⊂σ(A) and
density operatorρ one has
lim n→∞|Tr(ρµA(∆))−Tr(ρµAn(∆))|= 0. (7.11) Proof. Because |Tr(ρµA(∆))−Tr(ρµAn(∆)| ≤ X a∈∆ |Tr(ρµA({a}))−Tr(ρµAn({a})|, (7.12)
it suffices to show that (7.11) holds for singleton sets. By making use of Lemma A.5, it follows that
|Tr(ρµA({a}))−Tr(ρµAn({a})|=|Tr (ρ(µA({a})−µAn({a})))|
≤Tr(|ρ|)kµA({a})−µAn({a})k
=kµA({a})−µAn({a})k.
(7.13)
Now for everya∈σ(A) define the polynomial
pa(x) := Y a0∈σ(A) a06=a x−a0 a−a0. (7.14)
Restricted toσ(A), this function acts as a Kronecker delta function: pa(a0) =δaa0.
Therefore
kµA({a})−µAn({a})k=kpa(A)−pa(An)k. (7.15)
The finite-precision argument
7.3
The restriction that σ(A) =σ(An) for all nis a bit severe, and it is possible to
weaken it. However, doing so makes the theorem a lot messier6and the formulation given here suffices for the present discussion. The problem at hand is that, given an experimental procedure, one has to select a corresponding self-adjoint operator. Since the set of possible outcomes is determined by the experimental procedure, the choice is effectively between operators with the same spectrum. Now one may object that, because of the finite precision of measurements, one is not able to determine exactly what this spectrum should be. This is true of course, but this can be viewed as part of a separate selection process, where one can choose between different scalings of the spectrum on a collection of operators with the same spectrum. The selection of an operator is then split into the selection of a spectrum, and of a set of projection operators. This splitting can be justified by noting that scaling the spectrum does not alter the relevant probabilities (see also the discussion on page 38).
To make the finite precision argument work one has to show that for every observable A and every possible candidateA∈ Osa for representingA, there exists an operator A0 ∈ OMKC that is just as good a candidate as A given the finite
precision with which the properties of A can be measured. Here OMKC is the set
of operators corresponding to observables that can be assigned definite values by a hidden variable state in a non-contextual way. That is, for every possible function f :Obs → Osa there should be a functionf0 :Obs → OMKCsuch thatkf(A)−f0(A)k
is small for all A ∈Obs. It is a corollary of a theorem by Kent (1999) that such a set can indeed be constructed. Specifically, the following was shown.7
Theorem 7.3. For every finite-dimensional Hilbert space there exists a countable set of framesFMKC:={(Pin)|n∈N}that is dense in the set of all frames, i.e., for every frame (Pi) and every >0, there exists an n∈N such that
max
i minj kPi−P n
jk< . (7.16)
Furthermore, there exist functions
λ:LMKC1 (H)→ {0,1} (7.17) that satisfy
X
i
λ(Pin) = 1 ∀n∈N, (7.18)
6For example, it requires introducing a sequence of maps that link subsets ofσ(A) to subsets of
σ(An) and adjust (7.11) accordingly. Without such an adjustment the theorem would no longer hold as one can construct sequences (An)n∈Nsuch that limn→∞kA−Ank= 0 whileσ(A)∩σ(An) = 0 for alln.
7
Although it is implicitly assumed here that measurements can be represented by PVMs (as a consequence of OP), nothing hinges on this assumption. A similar theorem can be proven for POVMs.
where LMKC1 (H) is the set of all 1-dimensional projections that occur in one of the frames in FMKC.
Corollary 7.1. For every finite-dimensional Hilbert space there is a set of self- adjoint operatorsOMKC ⊂ Osa such that
(i) OMKC is closed under functional operations: for every real-valued Borel func- tion f, if A∈ OMKC, then f(A)∈ OMKC,
(ii) OMKC is dense in Osa: for every > 0 and every A0 ∈ Osa there is an A∈ OMKC such thatkA−A0k< ,
and there exist functionsλ:OMKC→R such that λ(A)∈σ(A),
λ(f(A)) =f(λ(A)) (7.19)
for everyA∈ OMKC and real-valued Borel function f.
Proof. Let {(Pin)|n∈N} be a set frames as in Theorem 7.3. For every n∈N let
Anbe the Abelian von Neumann algebra generated by the frame (Pin). Now define
OMKC:=
[
n∈N
(Osa∩ An). (7.20)
It immediately follows that criterion (i) is satisfied because the algebras are closed under the functional operations.
Now suppose A0 ∈ Osa and > 0 are given. Let (Pi) be a frame such that
A0 =P
ia0iPi. Now choose a frame (Pin) such thatkPi−Pink< dkAk for alli, where
dis the dimension of the Hilbert space, and defineA:=a0iPn
i . It follows that kA−A0k= X i a0i(Pin−Pi) ≤X i |a0i|kPin−Pik< . (7.21)
This establishes that (ii) holds.
Finally, every function λ that satisfies the criteria in Theorem 7.3 can be ex- tended to a functionλonOMKC that satisfies (7.19). To see this note that for every A∈ OMKC there exists a frame (Pin) such that A=
P iaiPin, withai ∈σ(A). Now define λ(A) :=X i aiλ(Pin). (7.22)
Then (7.19) follows from the spectral theorem together with the Borel functional calculus.
Recovering probabilities
7.4
The finite precision argument, together with the above theorem and corollary, motivate the following weakening of IP.
IPMKC (Meyer-Kent-Clifton Identification Principle)For every self-adjoint operatorA∈ OMKC there exists an observableA such thatAis associated
with A via OP.
It follows from Corollary 7.1 that QMKS, HV, NC and IPMKCcan be jointly satisfied.
In short, the result explicitly disproves the credo that the Kochen-Specker theorem establishes the impossibility of non-contextual hidden variable theories. On the more cautious side though, what has been established thus far is the possibility of non-contextual hidden variable states. What is still an open question at this point, is whether these states can be used to construct a classical representation of quantum probability. The answer is yes, but showing this requires a bit more work.