arXiv:1611.03826

Bell-like examples for spin-1 hidden variable theories

Kaushik Borah, N. D. Hari Dass

Published Nov 11, 2016• 0 citations

Abstract

Though John Bell had claimed that his spin-1/2 example of a hidden-variable theory(HV) is an \emph{explicit} counterexample to von Neumann's proof of the non-existence of hidden variable theories empirically equivalent to quantum mechanics, such examples can be so construed only if they met all of von Neumann's requirements. In particular, that they reproduced all the observed predictions of quantum theory. To shed light on these aspects, we have, on the one hand, simplified and critically examined Bell's original example and on the other hand, constructed explicit such examples for spin-1 systems. We have clarified the relation of our example to the Kochen-Specker and Bell's powerful earlier results. Our spin-1 examples are manifestly non-contextual, yet violating the K-S constraints configuration by configuration. Nevertheless, they reproduce the correct quantum expectation values and variances for arbitrary linear combinations of the beables in one case, and for close approximants to the beables representing K-S constraints. In conformity with the K-S theorem, the variance of the K-S constraint is nonzero. The implications of this non-vanishing variance are analysed in detail. In the other example, we show how this variance can be made arbitrarily small but not zero.

DOI: 10.48550/arxiv.1611.03826

Related papers

On a contextual model refuting Bell's theorem

Eugen Muchowski

Abstract Bell's theorem is refuted by presenting a counterexample that correctly predicts the expectation values of QM. As Bell only ruled out non-contextual models, a contextual model with hidden variables can refute his theorem. Such a model, which is able to explain the spin measurement results with entangled photons or electrons, is presented herein. It is not ruled out by the Kochen-Specker theorem. A physical justification for the contextual behaviour of entangled systems is provided. Consequences for the feasibility of quantum computers are discussed.

Europhysics Letters (EPL)Apr 1, 202111 citations

Hidden variable theories and quantum nonlocality

A. David. Boozer

We clarify the meaning of Bell's theorem and its implications for the construction of hidden variable theories by considering an example system consisting of two entangled spin-1/2 particles. Using this example, we present a simplified version of Bell's theorem and describe several hidden variable theories that agree with the predictions of quantum mechanics. These example theories clarify some subtle points, which are often misunderstood, regarding what it is that Bell's theorem actually establishes. \n

European Journal of PhysicsFeb 9, 20092 citations

Hidden-Variable Theories

Reinhold A. Bertlmann, Nicolai Friis

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

Sep 7, 20230 citations

On the Unconditional Validity of J. von Neumann's Proof of the Impossibility of Hidden Variables in Quantum Mechanics

C. S. Unnikrishnan

The impossibility of theories with hidden variables as an alternative and replacement for quantum mechanics was discussed by J. von Neumann in 1932. His proof was criticized as being logically circular, by Grete Hermann soon after, and as fundamentally flawed, by John Bell in 1964. Bell's severe criticism of Neumann's proof and the explicit (counter) example of a hidden variable model for the measurement of a quantum spin are considered by most researchers, though not all, as the definitive demonstration that Neumann's proof is inadequate. Despite being an argument of mathematical physics, an ambiguity of decision remains to this day. I show that Neumann's assumption of the linear additivity of the expectation values, even for incompatible (noncommuting) observables, is a necessary constraint related to the nature of observable physical variables and to the conservation laws. Therefore, any theory should necessarily obey it to qualify as a physically valid theory. Then, obviously, the hidden variable theories with dispersion-free ensembles that violate this assumption are ruled out. I show that it is Bell's counter-example that is fundamentally flawed, being inconsistent with the factual mechanics. Further, it is shown that the local hidden variable theories, for which the Bell's inequalities were derived, are grossly incompatible with the fundamental conservation laws. I identify the intrinsic uncertainty in the action as the reason for the irreducible dispersion, which implies that there are no dispersion-free ensembles at any scale of mechanics. With the unconditional validity of its central assumption shown, Neumann's proof is fully resurrected.

arXiv:2105.13996May 27, 20210 citations

Kochen-Specker Theorem and Games

Laura Mančinska

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:

Jan 1, 20070 citations

Hidden Variables, Non Contextuality and Einstein-Locality in Quantum Mechanics

Virendra Singh

We discuss the problem of hidden variables and the motivation for introducting them in quantum mechanics. These include determinism, and the problem of meassurement and incompleteness. We first discuss Von-Neumann's imposisbility proof and then analyse it's weakness in terms of Bell's explicit hidden variable model of spin one-half particles. We next discuss Gleason's theorem and Kochen-Specker theorem and bring out the troublems with non contextual hidden variable theories. An important role is played by Einstein locality in the discussion of hidden variable theories as was first brought out by Einstein, Podolsky and Rosen. We elaborate it's various implications such as Bell's theorem in terms of Bell's inequalities as well as later work in which Bell's theorem follows without using inequalities.

arXiv:quant-ph/0507182Jul 19, 20051 citation