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MOJ Ecology & Environmental Sciences

The Kochen-Specker Theorem with Two Trials of Measurements

Koji Nagata

Published Mar 13, 2017• 0 citations
Publisher

Abstract

We review non-classicality of quantum datum.We consider whether we can assign the predetermined "hidden" result to numbers 1 and -1 as in results of measurements in a thought experiment.We assume the number of measurements is two.If we detect 1 / 2 as 1 and detect | ↓ as -1, then we can derive the Kochen-Specker theorem.The same situation occurs when we use a new measurement theory that the results of measurements are either 1 / 2 or -1 / 2

DOI
10.15406/mojes.2017.02.00011

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arXiv:1506.04150•Physical Review Letters•Sep 9, 2015•87 citations
We review non-classicality of quantum datum.We consider whether we can assign the predetermined "hidden" result to numbers 1 and -1 as in results of measurements in a thought experiment.We assume the number of measurements is two.If we detect 1 / 2 as 1 and detect | ↓ as -1, then we can derive the Kochen-Specker theorem.The same situation occurs when we use a new measurement theory that the results of measurements are either 1 / 2 or -1 / 2
We review non-classicality of quantum datum. We consider whether we can assign the predetermined ^^ ^^ hidden'' result to numbers 1 and −1-1−1 as in results of measurements in a thought experiment. We assume the number of measurements is two. If we detect ∣↑⟩|\uparrow\rangle∣↑⟩ as 1 and detect ∣↓⟩|\downarrow\rangle∣↓⟩ as −1-1−1, then we can derive the Kochen-Speker theorem. The same situation occurs when we use a finite-precision measurement theory that the results of measurements are either 1−ϵ1-\epsilon1−ϵ or −1+ϵ-1+\epsilon−1+ϵ.
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:
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.
We consider the two quantum measurement theories for measuring a single Pauli observable. We assume also the existence of a classical probability space for the two measurement theories. We cannot avoid the Kochen-Specker (KS) contradiction when we measure the Pauli observable by using the projective measurement theory if we introduce a classical probability space. The results of measurement are either +1+1+1 or −1-1−1 (in ℏ/2\hbar/2ℏ/2 unit) when we consider a spin-1/2 system. The projective measurement theory does not accept a classical probability space when we measure the Pauli observable. We propose a new measurement theory based on the truth values, i.e., the truth T (1) for true and the falsity F (0) for false. The results of measurement are either +1+1+1 or 0 (in ℏ/2\hbar/2ℏ/2 unit). We avoid the KS contradiction when we measure the Pauli observable by using the new measurement theory if we introduce a classical probability space. The new measurement theory accepts a classical probability space when we measure the Pauli observable.
We derive new type of no-hidden-variables theorem based on the assumptions proposed by Kochen and Specker. We consider NNN spin-1/2 systems. The hidden results of measurement are either +1+1+1 or −1-1−1 (in ℏ/2\hbar/2ℏ/2 unit). We derive some proposition concerning a quantum expected value under an assumption about the existence of the Bloch sphere in NNN spin-1/2 systems. However, the hidden variables theory violates the proposition with a magnitude that grows exponentially with the number of particles. Therefore, we have to give up either the existence of the Bloch sphere or the hidden variables theory. Also we discuss two-dimensional no-hidden-variables theorem of the KS type. Especially, we systematically describe our assertion based on more mathematical analysis using raw data in a thoughtful experiment.
The Kochen-Specker theorem demonstrates that it is not possible to reproduce the predictions of quantum theory in terms of a hidden variable model where the hidden variables assign a value to every projector deterministically and noncontextually. A noncontextual value assignment to a projector is one that does not depend on which other projectors-the context-are measured together with it. Using a generalization of the notion of noncontextuality that applies to both measurements and preparations, we propose a scheme for deriving inequalities that test whether a given set of experimental statistics is consistent with a noncontextual model. Unlike previous inequalities inspired by the Kochen-Specker theorem, we do not assume that the value assignments are deterministic and therefore in the face of a violation of our inequality, the possibility of salvaging noncontextuality by abandoning determinism is no longer an option. Our approach is operational in the sense that it does not presume quantum theory: a violation of our inequality implies the impossibility of a noncontextual model for any operational theory that can account for the experimental observations, including any successor to quantum theory.