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Philosophy of Science

The Sum Rule is Well-Confirmed

Clark Glymour

Published Mar 1, 1977• 6 citations
Publisher

Abstract

Simon Kochen and Ernst Specker's well-known argument ([6]) against hidden variable theories for quantum mechanics is also an argument against the possibility of quantum systems having, simultaneously, precise values for all of the dynamical quantities associated with such systems. Devices for defeating the argument were in the literature even before its publication, but recently Arthur Fine has raised a new difficulty ([4]). Fine points out that Kochen and Specker's argument requires the following principles: Sum Rule: At all times, in all states, the value of the quantity with operator A + B is the sum of the values of the quantity with operator A and the quantity with operator B, provided A, B commute. Product Rule: At all times, in all states, the value of the quantity with operator AB is the product of the values of the quantities with operators A and B, provided A, B commute.

DOI
10.1086/288725

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Simon Kochen and Ernst Specker's well-known argument ([6]) against hidden variable theories for quantum mechanics is also an argument against the possibility of quantum systems having, simultaneously, precise values for all of the dynamical quantities associated with such systems. Devices for defeating the argument were in the literature even before its publication, but recently Arthur Fine has raised a new difficulty ([4]). Fine points out that Kochen and Specker's argument requires the following principles: Sum Rule: At all times, in all states, the value of the quantity with operator A + B is the sum of the values of the quantity with operator A and the quantity with operator B, provided A, B commute. Product Rule: At all times, in all states, the value of the quantity with operator AB is the product of the values of the quantities with operators A and B, provided A, B commute.
We use the validity of Addition and Multiplication for a hidden variables theory. First, we provide an example that the two operations Addition and Multiplication do not commute with each other as revealed by the analyses that are performed in a finite set of numbers. Our discussion leads to an initial conclusion that Sum rule and Product rule do not commute with each other in a hidden variables theory. If we accept this conclusion, we do not get the Bell- Kochen-Specker paradox. In more detail, quantum mechanics may accept the hidden variables theory. Next, we discuss the validity of operators under an assumption that Sum rule and Product rule commute with each other. In this case, we indeed get the Bell- Kochen-Specker paradox. We got the non-classicality of macroscopic experimental data observed in the Stern-Gerlach experiment and the double-slit experiment. If we detect | ↑> and then we detect | ↓>, the experiments cannot accept the hidden variables theory. We considered whether we can assign the predetermined “hidden ” result to numbers 1 and-1 as in results of measurements with the number of measurements finite (e.g., twice) in the experiments. It turned out that we cannot assign the predetermined hidden result to such results of measurements. The next conclusion indicates interestingly that the Stern-Gerlach experiment cannot accept classical mechanics. The double-slit experiment had led to the same situation, and they were indeed quantum mechanical phenomena.
Valuation functions of observables in quantum mechanics are often expected to obey two constraints called the sum rule and product rule. However, the Kochen–Specker (KS) theorem shows that for a Hilbert space of quantum mechanics of dimension d ≤ 3, these constraints contradict individually with the assumption of value definiteness. The two rules are not irrelated and Peres [Found. Phys. 26 (1996) 807] has conceived a method of converting the product rule into a sum rule for the case of two qubits. Here we apply this method to a proof provided by Mermin based on the product rule for a three-qubit system involving nine operators. We provide the conversion of this proof to one based on sum rule involving ten operators.
In this talk I present a simple derivation of an old result of Kochen and Specker, which is apparently unrelated to the famous work of Bell on hidden variables, but is presumably equally important. Kochen and Specker showed in 1967 that quantum mechanics cannot be embedded into a classical stochastic theory, provided the quantum theoretical probability distributions are reproduced and one additional highly desirable property is satisfied. This showed in a striking manner what were the difficulties in implementing the Einstein programme of a `complete' version of quantum mechanics.
One of the statements one would hear pretty early in any quantum mechanics course is that for quantum particles there are properties which do not have definite values. For example, if we measure |+〉 in standard basis, the best we can do to describe the outcome is to say, that with probability 1/2 we will get 0 and with probability 1/2 we will get 1. Yet one could wonder, is this really the way things are or is it just because our description of this quantum particle is incomplete, but in fact there are some hidden variables inside every quantum system that contain enough information to tell the outcome of measurement (see Fig. 1). However, many people disdain the idea of hidden variables, so there has been rather long history of theorems that forbid theories involving hidden variables. Usually if one wants to prove some no hidden variables theorem he has to assume something about the hidden variables theories he is going to forbid. One of such assumptions is non-contextuality, which means that the outcome we get after measuring with an observable is the same regardless of what other observables we choose to measure at the same time. The first theorem forbidding hidden variables theories is due to von Neumann in 1932. Then for a third of a century everyone cited von Neumann’s work carelessly, till Bell in 1966 indicated that there was an unreasonable assumption in von Neumann’s proof and urged for further examination of the problem (see [6] and [1] for more historical details). Bell also constructed a hidden variables model for one qubit that could assign the outcome of any observable in a way consistent with quantum mechanics. He also proved two theorems that forbid non-contextual hidden variables theories using counterexamples with larger quantum systems. Yet surprisingly he himself thought that hidden variables do exists and considered these theorems only as characterizations of the hidden variables theory we should look for. In this essay we will discuss several versions of one of the two Bell’s theorems. The theorem we will consider was also independently proven by Kochen and Specker in 1967 (see [5]), hence it is called Bell-Kochen-Specker theorem. For shortness and in order not to mix this theorem with another Bell’s theorem from now on we will refer to it as Kochen-Specker theorem. We will be interested in proving several versions of this theorem and the following lemma will turn out to be of great use: