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Quantum Mechanics and No-Hidden Variable Theories: Comparing the Bell, Kochen-Specker and Free Will Theorems

P.A. Steijlen

Published Aug 26, 2014• 0 citations
OpenAlex

Abstract

This thesis explores the exact differences between the two most well-known no-hidden variable theories: the Bell theorem and the Kochen-Specker theorem. Furthermore, it tries to place a newer result called `the free will theorem', as developed by John Conway and Simon Kochen, in this context. To compare the Bell and Kochen-Specker theorems a reformulation of the two theorems by David Mermin will be discussed, which turns out to lead to an even stronger no-hidden variable theory. From the discussion of the free will theorem and the critical reactions it has elicited, it will become clear that the free will theorem is actually very similar to this stronger no-hidden variable theory.

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This thesis explores the exact differences between the two most well-known no-hidden variable theories: the Bell theorem and the Kochen-Specker theorem. Furthermore, it tries to place a newer result called `the free will theorem', as developed by John Conway and Simon Kochen, in this context. To compare the Bell and Kochen-Specker theorems a reformulation of the two theorems by David Mermin will be discussed, which turns out to lead to an even stronger no-hidden variable theory. From the discussion of the free will theorem and the critical reactions it has elicited, it will become clear that the free will theorem is actually very similar to this stronger no-hidden variable theory.
Abstract In chapter 12 we give an overview of the historical development of hidden-variable theories aiming to attribute ‘hidden’ definite outcomes to any potential measurement on a quantum system. We start with a discussion of von Neumann’s assumption on hidden-variable models, which was later realized not to hold by Hermann, Kochen, Specker, and Bell, going into detail on Bell’s refutation of von Neumann’s arguments. We then turn to the notion of contextuality, which severely restricts hidden-variable models: we discuss Gleason’s theorem before stating and proving the famous Kochen-Specker theorem, as well as the simpler versions of Peres’ nonet and Mermin’s pentagram. These results show that quantum mechanics is not compatible with the assignment of definite measurement outcomes to all possible measurements, and such assignments can hence only be made once a restricted set of observables, a context, is selected
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.
In this work we begin with a critical review of the concept of observable in a generical physical theory; relying on C*-algebras formalism and on Gelfand-Naimark theorem we justify the link between observables and operators in an Hilbert space. We discuss particularly observables of quantum mechanics, introducing some mathematical results that characterizes them. We introduce the problem of hidden variables and the hypothesis of non-contextuality, than we give a proof of the Kochen-Specker theorem in dimension greater or equal to 3 using partial algebras formalism; we also show a constructive example of an hidden variable theory in dimension 2. Finally we discuss Meyer’s and Kent’s objections relying on finite precision of physical measurements and we present colorations of S² ∩ Q³ and of dense subsets of the projectors of C^n and l²(C) that respect Kochen-Specker rules.
Modal quantum theory (MQT) is a simplified cousin of ordinary Hilbert space quantum theory. We show that two important theorems of actual quantum theory, the Kochen-Specker theorem excluding non-contextual hidden variables and the Conway-Kochen "free will theorem" about entangled systems, have direct analogues in MQT. The proofs of these analogue theorems are similar to, but much simpler than, the originals. We also show that the structure of possible measurement results for an entangled system in MQT cannot be represented by probability assignments satisfying the no-signaling principle, such as those given by ordinary quantum theory.
Abstract In this contribution 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.