Although the Kochen-Specker theorem purports to make a contribution to the phi- losophy of quantum mechanics, its precise importance is not immediately clear. This is a necessary consequence for any formal proof in philosophy; at some point philosophical intuitions have to be translated into mathematically well-defined as- sumptions, and derived consequences then have to be translated back. These trans- lations are always somewhat muddy. However, in the present situation there is room for improvement. In the previous section it was not always clear where the philosophy stopped and the mathematics began. In this section I will make the separation more precise.
Since the mathematics is the easiest part I start there. Specifically, I start with a purely mathematical reformulation of the theorem. The mathematical assumptions are then scrutinized and motivated/derived from more conceptual/philosophical as- sumptions. This then allows the reformulation of the theorem in a more conceptual
1
Actually, in the finite-dimensional case every function can be shown to behave like a polynomial. That is, for every functionf and every self-adjoint operatorA, there is a polynomial pf,Asuch thatf(A) =pf,A(A).
Reformulating the Kochen-Specker theorem
6.2
language. In further sections then, the conceptual assumptions are subjected to a more intensive investigation. Without further ado, here is the Kochen-Specker theorem (again).
Theorem 6.2. Let H be a Hilbert space of dimension at least 3, and letOsa be the
set of self-adjoint operators onH. Then there is no function λ:Osa →Rsuch that
λ(A)∈σ(A), (6.16a) λ(f(A)) =f(λ(A)) (6.16b)
for everyA∈ Osa and real-valued Borel functionf.
This presentation of the theorem can be understood as a corollary of Gleason’s theorem, which implies the non-existence of 0-1 valued quantum probability func- tions. On the other hand, anyλsatisfying the constraints (6.16) gives rise to a 0-1 valued probability function by restriction to the projection operators. The impor- tant distinction is conceptual in nature.2 Gleason proved his theorem as part of a purely mathematical investigation (i.e., the classification of all probability func- tions). Kochen and Specker, on the other hand, intended their theorem to be philosophically relevant by relating it to the question of the possibility of hidden variables. Such relevance can only be had by establishing relations between the Hilbert space structure, the empirical content of quantum mechanics, and further philosophical desiderata. So let’s draw some relations.
In Theorem 6.2 the function λis intended to represent a hidden variable state. The interest in the existence of such states comes from a desire to find a description of quantum mechanics that is more complete than the standard formalism. The standard formalism is silent about the origin of measurement outcomes; it is not clear whether they can be thought of as revelations of quantities existing indepen- dent of the performance of a measurement. Hidden variables are supposed to fill in this gap.3 Formally, postulating their existence can be seen to be embodied by the conjunction of the following two assumptions.
VD (Value Definiteness) Every observable possesses a unique definite value at all times.
FM (Faithful Measurement)Given VD, a measurement of an observable reveals the value it possesses at the time of the measurement.
2
To be fair, the original Kochen-Specker theorem is not a corollary of Gleason’s theorem as they do not assume thatλ(A) is defined for allA∈ Osa. So apart from the conceptual distinction
between the two theorems there is also a mathematical one, which is discussed in chapter 7. However, to clarify the conceptual distinction the formulation in Theorem 6.2 suffices.
3There may be other reasons for wishing to look for hidden variable formulations of quantum
One may note a discrepancy between these assumptions and the role of the functionλin Theorem 6.2. VD and FM refer to a notion of hidden variable states as given by Definition 5.2. Such a hidden variable state acts on the set of observables. The functionλon the other hand is defined on the set of self-adjoint operators. The connection between the two is made by the observable postulate.
OP (Observable Postulate)Every observable A for the system S is associated with a self-adjoint operatorA acting onH.
While it is clear that every function λsatisfying (6.16) induces a hidden variable state by assigning to each observable A the value λ(A) (where A is the operator associated with A by OP), it does not follow from this same postulate that every hidden variable state should be of this form. It would follow if one could take the set of observables to beidentical to the set of self-adjoint operators. Two further assumptions can be recognized that together establish this.
NC (Non-Contexuality)Every self-adjoint operator is associated with at most one observable.
IP (Identification Principle) For every self-adjoint operator A there exists an observableA such that Ais associated with A via OP.
To illustrate how these assumptions all tie together, note that OP implies the existence of a functionf :Obs → Osa that assigns to each observable a self-adjoint
operator. NC then states that f is injective, and IP states that it is surjective. Consequently, any hidden variable stateκ:Obs →Rcan be translated to a function λ:Osa→R without loss of generality.
Thus far I have only discussed the domain of hidden variable states, but not their range or possible constraints they should satisfy. The first of the two constraints (6.16a) (together with the earlier assumptions) ensures thatλ is a hidden variable state as in Definition 5.2. But the motivation for this constraint comes of course from quantum mechanics itself. In fact, it can be seen to follow from FM and EFR by making use of Example 6.1.
The second constraint (6.16b) is the FUNC rule from the previous section. The empirical motivation for this constraint derives from EFR. But EFR cannot do any work without a further assumption on the comeasurability of observables. CoP suffices to make this final step, but a slightly weaker version also works. Although weakening assumptions is often a good idea to obtain stronger results, the concep- tual motivation for this particular weakening is lacking at this point. The explana- tion will be given in the next section, where it turns out to be the case that this slight alteration is quite crucial.
Reformulating the Kochen-Specker theorem
6.2
WCoP (Weak Comeasurability Postulate) For every finite set of observ- ables{A1, . . . ,An} with corresponding self-adjoint operatorsA1, . . . , An, if
[Ai, Aj] = 0 for alli, j, then there exists a set of observables{A10, . . . ,An0},
corresponding to the same operators, such that it is possible to perform a joint measurement of A10, . . . ,An0.
The FUNC rule can now be seen to follow from WCoP and EFR together with the other assumptions. This establishes a solid empirical underpinning of one of the more technical ingredients of the Kochen-Specker theorem. To sum up, the theorem can be formulated as follows.
Theorem 6.3. For any system described by a Hilbert space with dimension greater than 2 the assumptions OP, WCoP, EFR, VD, FM, IP and NC taken together lead to a contradiction.
The implications of the theorem are straightforward: at least one of the as- sumptions has to be rejected.4 The assumptions OP, WCoP, EFR are all com- monly accepted facets of quantum mechanics and are to be kept in place to keep the discussion on track.5 For further exposition, it is then useful to define
QMKS:= OP∧WCoP∧EFR (6.17)
as the part of quantum mechanics used in the Kochen-Specker theorem. VD and FM are reasonable assumptions for a hidden variable theory, and it is likewise useful to set
HV := VD∧FM. (6.18)
Also, these are requirements for the possibility of constructing a classical probability space for quantum mechanics (see section 5.2). Theorem 6.3 can then be formulated as the formula
QMKS∧HV∧IP∧NC⇒ ⊥. (6.19) In this formulation it is clear that the Kochen-Specker theorem relies on two as- sumptions that lack a good motivation: IP and NC. When adopting a purely formal stance with respect to the question of classical representations of quantum probabil- ity these assumptions have a certain natural appeal. Rejecting IP would imply that not every quantum random variable (self-adjoint operator) will be represented by at least one classical random variable. Not all facets of quantum probability would then be captured by the classical representation. Rejecting NC would imply that the quantum random variables do not fully take into account all relevant random variables, as some of them refer to multiple distinct classical random variables. If
4Unless one is willing to resort to paraconsistent logic. 5
this is the case, it raises the question of how the quantum formalism was found to be adequate in the first place. One would expect that in some way a distinction between random variables is able to represent itself. These considerations of course do not pose a definitive argument against the view that quantum probability cannot have a satisfactory classical representation. However, they do indicate that there are difficulties for a ‘natural’ classical representation.
For the question of hidden variables the costs of rejecting either IP or NC may appear to be less high. This would especially be the case when the rewards can be as big as solving the measurement problem. But there are benefits closer to home. Rejecting NC may be quite an attractive option. The incompleteness of the quantum formalism implied by denying it (Osa does not adequately describe all observables) can even be seen as a bonus. Indeed, the incompleteness of quantum mechanics was the main motivation for going into the question of hidden variables in the first place. The Kochen-Specker theorem then only confirms the idea that something is missing.
Rejecting IP is more complicated. To be sure, there are good reasons to reject its validity. A nice example is given by Nielsen (1997) who constructs a self-adjoint operator that contradicts the Church-Turing thesis if it were to correspond to an observable. And as noted in chapter 3 there are many operators for which it is hard to imagine any experimental setup that would correspond to it. But it is difficult to find a satisfactory argument to ban a specific set of operators from the observables, and to show that by doing so the Kochen-Specker theorem can be circumvented. This is particularly difficult when realizing that the original proof of Kochen and Specker uses a much weaker version of IP, requiring only that a particular finite set of operators correspond to observables. In short, trying to circumvent their theorem by wiggling with IP is quite non-trivial and thus deserves to be discussed in a separate chapter.
Here I have only summed up some intuitions one may have considering the implications of the Kochen-Specker theorem. But real insight can only be gained by having a closer look at the assumptions IP and NC. The rejection of NC is discussed in the next section and the rejection of IP is the topic of chapter 7.