7/28/2019 The Kochen-Specker Theorem (4) - The Functional Composition Principle

1/2

The Functional Composition Principle

The key ingredients of the KS theorem are the constraints on value assignments spelled out in (2): the Sum Rule and Product Rule. They can be derived from a more general principle, called the Functional Composition Principle (FUNC).[11] Theprinciple trades on the mathematical fact that for a self-adjoint operator A operating on a Hilbert space, and an arbitrary function f: RR (where R is the set of the real numbers), we can define f(A) and show that it also is a self-adjointoperator (hence, we write f(A)). If we further assume that to every self-adjointoperator there corresponds a QM observable, then the principle can be formulated thus:

FUNC: Let A be a self-adjoint operator associated with observable A, let f:R R be an arbitrary function, such that f(A) is another self-adjoint operator,and let |> be an arbitrary state; then f(A) is associated uniquely with an observable f(A) such that:

v(f(A))|> = f(v(A))|>

(We introduce the state superscript above to allow for a possible dependence ofvalues on the particular quantum state the system is prepared in.) The Sum Ruleand the Product Rule are straightforward consequences of FUNC [Proof]. FUNC itself is not derivable from the formalism of QM, but a statistical version of it (called STAT FUNC) is [Proof]:

STAT FUNC: Given A, f, |> as defined in FUNC, then, for an arbitrary real number b:

prob[v(f(A))|>=b] = prob[f(v(A))|>=b]

But STAT FUNC cannot only be derived from the QM formalism; it also follows fromFUNC [Proof]. This can be seen as providing a plausibility argument for FUNC (Redhead 1987: 132): STAT FUNC is true, as a matter of the mathematics of QM. Now, if FUNC were true, we could derive STAT FUNC, and thus understand part of the mathematics of QM as a consequence of FUNC.[12]

But how can we derive FUNC itself, if not from STAT FUNC? It is a direct consequ

ence of STAT FUNC and three assumptions (two of which are familiar from the introduction):

Value Realism (VR): If there is an operationally defined real number , associated with a self-adjoint operator A and if, for a given state, the statistical algorithm of QM for A yields a real number with = prob(v(A)=), then there exists an observable A with value .

Value Definiteness (VD): All observables defined for a QM system have definite values at all times.

Noncontextuality (NC): If a QM system possesses a property (value of an observable), then it does so independently of any measurement context.

VR and NC require further explanation. First, we need to explain the content ofVR. The statistical algorithm of QM tells us how to calculate a probability froma given state, a given observable and its possible value. Here we understand itas a mere mathematical device without any physical interpretation: Given a Hilbert space vector, an operator and its eigenvalues, the algorithm tells us how tocalculate new numbers (which have the properties of probabilities). In addition, by operationally defined we here simply mean made up from a number which we knowto denote a real property. So, VR, in effect, says that, if we have a real property (value of an observable G), and we are able to construct from a new number a

7/28/2019 The Kochen-Specker Theorem (4) - The Functional Composition Principle

2/2

d find an operator A such that is an eigenvalue of A, then (we have fulfilled everything necessary to apply the statistical algorithm; thus) A represents an observable A and its value is a real property.

Second, a failure of NC could be understood in two ways. Either the value of anobservable might be context-dependent, although the observable itself is not; orthe value of an observable might be context-dependent, because the observable itself is. In either case, the independence from context of an observable impliesthat there is a correspondence of observables and operators. This implication of NC is what we will use presently in the derivation of FUNC. We will indeed assume that, if NC holds, this means that the observable and thereby also its value is independent of the measurement context, i.e. is independent of how it is measured. In particular, the independence from context of an observable implies that there is a 1:1 correspondence of observables and operators. This implication of NC is what we will use presently in the derivation of FUNC. Conversely, failure of NC will be construed solely as failure of the 1:1 correspondence.

From VR, VD, NC and STAT FUNC, we can derive FUNC as follows. Consider an arbitrary state of a system and an arbitrary observable Q. By VD, Q possesses a valuev(Q)=a. Thus, we can form the number f(v(Q))=b for an arbitrary function f. Forthis number, by STAT FUNC, prob[f(v(Q))=b] = prob[v(f(Q))=b]. Hence, we have, bytransforming probabilities according to STAT FUNC, created a new self-adjoint operator f(Q), and associated it with the two real numbers b and prob[f(v(Q))=b].Thus, by VR, there is an observable corresponding to f(Q) with value b, hence f

(v(Q))=v(f(Q)). By NC, that observable is unique, hence FUNC follows.