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Journal of Mathematical Physics

Violation of the Quantum Ordering of Propositions in Hidden-Variable Theories

James Turner

Published Sep 1, 1968• 7 citations
Publisher

Abstract

A general definition of hidden-variable theories in terms of the dual structure of states and propositions is proposed. As a consequence of a theorem due to Zierler and Schlessinger, this definition implies a violation of the quantum ordering of propositions in the corresponding hidden-variable theory. This violation is shown explicitly for the theory of measurement due to Bohm and Bub.

DOI
10.1063/1.1664730

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A general definition of hidden-variable theories in terms of the dual structure of states and propositions is proposed. As a consequence of a theorem due to Zierler and Schlessinger, this definition implies a violation of the quantum ordering of propositions in the corresponding hidden-variable theory. This violation is shown explicitly for the theory of measurement due to Bohm and Bub.
An abstract definition of a general hidden-variables theory is given, and it is shown that such a theory is always possible in the present framework of quantum mechanics and is, in fact, unique in a certain sense. It is noted that the Bohm-Bub hidden-variables example is contained in this theory and an attempt is made to clarify the position of this theory with respect to hidden-variable impossibility proofs. The general definition is used in the consideration of quantum-mechanical ordering and the measurement process.
A recently proposed theory of measurement for nonrelativistic quantum mechanics, due to Bohm and Bub, is generalized, and a mathematical analysis of the theory is presented. The fact that there are always theories of the Bohm-Bub type, given the standard formulation of quantum mechanics, is discussed. The phenomenological coefficient $\ensuremath{\gamma}$ of the original theory is defined to be the absolute value of the energy change of the system during measurement divided by $\ensuremath{\hbar}$. This definition leads to a Heisenberg-like relation between the average energy change of the system during the measurement process and the collapse time, but the new relation depends on the so-called hidden variables, and hence could lead to non-quantum-mechanical effects. The time dependence of the hidden variables is considered, and they are then assumed to be constant for each individual system. An experiment is proposed which would test this assumption. A positive result would disagree significantly with the quantum-mechanical prediction. The question of whether or not the new variables must remain unobservable is analyzed mathematically. A rigorous definition of simultaneous measurement is made, and the resulting interference is given a mathematical description.
We formulate stochastic noncontextual hidden-variable theories. In such theories the hidden state \ensuremath{\lambda} specifies not definite values, but expectation values A\ifmmode\bar\else\textasciimacron\fi{}(\ensuremath{\lambda}), . . . of observables A, . . . and noncontextuality means that A\ifmmode\bar\else\textasciimacron\fi{}(\ensuremath{\lambda}) is independent of which other commuting observables commuting with A are measured together with A. We show via Bell inequalities that such theories conflict with quantum theory and propose a two-photon experimental test. We also show that for a single particle of spin (2n{2}^{\mathit{n}}2n-1)/2, quantum violation of classical noncontextuality grows exponentially with n.
Abstract We will first define what is meant by “hidden variables”. Then, we will review theorems proving the impossibility of theories introducing such variables in a naive way. Next, we will outline the de Broglie-Bohm theory and show that it is not refuted by those theorems. To do that, we will explain how “measurements” work in that theory, using the examples of spin measurements and of momentum measurements. Those measurements do not reveal pre-existing properties of the quantum particle but are the result of interactions between that particle and the measuring apparatus. And this is why the de Broglie-Bohm theory is not refuted by the theorems on the impossibility of hidden variables. We will also explain the relationship between the no “hidden variables” theorems and nonlocality, with and without introducing Bell’s inequalities.
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
Quantum correlations that violate the Bell inequality cannot be explained by any local-hidden-variable theory that is measurement independent. However, this violation merely signifies the incompatibility of the underlying assumptions of reality, locality, and measurement independence, without providing a quantitative measure of the extent to which each assumption is violated. In contrast, Hall introduced a measure for each of the following assumptions: indeterminism, signaling, and measurement dependence, and Hall also generalized the Bell-Clauser-Horne-Shimony-Holt inequality, which provides a quantitative trade-off relationship between these assumptions. In this paper, we consider the introduction of hidden variables to be an essential assumption of Bell's theorem and introduce a quantification of hidden variables, which we term ``hiddenness.'' We derive a trade-off relation between hiddenness and measurement dependence that applies to any local-hidden-variable theory.