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Cambridge University Press eBooks

Existence of hidden variables having only upper probabilities

Patrick Suppes, Mario Zanotti

Published Nov 13, 1996• 0 citations
Publisher

Abstract

Great attention has been given to the exact formulation of locality conditions in the existence of hidden variables for quantum mechanical configurations that do not satisfy Bell's inequalities. In contrast, almost no attention has been given to examining what happens if the requirements for a probability measure are relaxed. The purpose of this paper is to show that local hidden variables do exist if the requirement of a probability measure is weakened to that of having only an upper probability measure. In the past several decades, upper and lower measures have been brought into the theory of statistical inference and the measurement of beliefs rather extensively, but in almost all cases considered in statistical theory or the belief framework, it is assumed that a probability measure exists, or in many cases, that an entire family of probability measures is given such that the sup and inf of the family defines the upper and lower measures. Sometimes still more is required, namely, that the upper–lower measure be a capacity (in the sense of Choquet) of infinite order. This is, for example, true of Dempster's theory of statistical inference. For an extensive recent analysis of the various theories of imprecise statistical models or belief functions using upper and lower probabilities, Walley (1991) is an excellent reference. In contrast and as is well known, in standard quantum mechanical cases the Bell inequalities cannot be satisfied and this implies that there exists no probability measure.

DOI
10.1017/cbo9781139172639.013

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Great attention has been given to the exact formulation of locality conditions in the existence of hidden variables for quantum mechanical configurations that do not satisfy Bell's inequalities. In contrast, almost no attention has been given to examining what happens if the requirements for a probability measure are relaxed. The purpose of this paper is to show that local hidden variables do exist if the requirement of a probability measure is weakened to that of having only an upper probability measure. In the past several decades, upper and lower measures have been brought into the theory of statistical inference and the measurement of beliefs rather extensively, but in almost all cases considered in statistical theory or the belief framework, it is assumed that a probability measure exists, or in many cases, that an entire family of probability measures is given such that the sup and inf of the family defines the upper and lower measures. Sometimes still more is required, namely, that the upper–lower measure be a capacity (in the sense of Choquet) of infinite order. This is, for example, true of Dempster's theory of statistical inference. For an extensive recent analysis of the various theories of imprecise statistical models or belief functions using upper and lower probabilities, Walley (1991) is an excellent reference. In contrast and as is well known, in standard quantum mechanical cases the Bell inequalities cannot be satisfied and this implies that there exists no probability measure.
The conceptually puzzling features of quantum mechanics as a statistical theory all have their source in the impossibility of relating the probability assignments defined by the quantum state to distributions over determinate values of the physical magnitudes. While some interpretations of quantum mechanics, as well as “hidden variable” modifications of the theory, have proposed several constructions for assigning values to magnitudes (beyond the value assignments defined by the quantum state alone), a variety of theorems exist which impose severe restrictions on such theoretical constructions. This note will deal with Bell's proof (in [2]), that any hidden variable theory satisfying a physically reasonable locality condition is characterized by an inequality which is inconsistent with the quantum statistics. Bell's result has initiated a series of experiments designed to refute the entire class of local hidden variable theories. We show that Bell's inequality actually characterizes a feature of hidden variable theories which is much weaker than locality in the sense considered physically motivated.
INTRODUCTION Fine (1982) proved that Bell's (1964) inequalities are sufficient, as well as necessary, to guarantee the existence of a joint probability distribution of the random variables in question, e.g., those representing certain spin correlation experiments. Garg and Mermin (1982) gave an example of eight random variables satisfying Bell's inequalities but such that no joint distribution exists. The purpose of this paper is to extend Bell's inequalities to obtain some general necessary conditions for the existence of a joint probability distribution for any finite collection of Bell-type random variables. Our results show that, for N > 4, many new elementary inequalities beyond those of Bell must be satisfied by any hidden-variable theory. Since these additional inequalities are violated by quantum mechanical predictions for appropriate choice of measurement arrangements, they serve to increase the conceptual distance between what may be called Einstein locality, after the EPR paradox, and quantum mechanics. To give a concrete sense of the nature of our results, we exhibit one of the new inequalities violated by Garg and Mermin's example. For theoretical purposes it will suffice to introduce Bell-type random variables X 1 ,…, X N , N ≥ 4, having possible values ±1, with means E ( X i ) = 0 and with what we term Bell covariances E ( X i X j ) relative to the index i 0 , with 1 ≤ i ≤ i 0 , N , 2 ≤ i 0 N – 2. Notice that, when N = 4, we must, as is familiar, have i 0 = 2.
It has been believed that statistical inequalities such as the Bell inequality should be modified once measurement independence (MI), the assumption that observers can freely choose measurement settings without changing the probability distribution of hidden variables, is relaxed. However, we show that there exists the possibility that the Bell inequality is still valid even if MI is relaxed. MI is only a sufficient condition to derive the Bell inequality when both determinism and setting independence, usually called local realism, are satisfied. We thus propose a condition necessary and sufficient for deriving the Bell inequality, called concealed measurement dependence.
We propose Bell inequalities for discrete or continuous quantum systems which test the compatibility of quantum physics with an interpretation in terms of deterministic hidden-variable theories. The wave function collapse that occurs in a sequence of quantum measurements enters the upper bound via the concept of quantum conditional probabilities. The resulting hidden-variable inequality is applicable to an arbitrary observable that is decomposable into a weighted sum of noncommuting projectors. We present local and nonlocal examples of violation of generalized Bell inequalities in phase space, which sense the negativity of the Wigner function.
It has been believed that statistical inequality such as Bell inequality should be modified once measurement independence (MI), the assumption that observers can freely choose measurement settings without changing the probability distribution of hidden variables, is relaxed. However, we show that there exists the possibility that Bell inequality is still valid even if MI is relaxed. MI is only a sufficient condition to derive Bell inequality when both determinism and setting independence, usually called as local realism, are satisfied. We thus propose a new condition necessary and sufficient for deriving Bell inequality, called as concealed measurement dependence.
Abstract The experimentally verified violation of Bell’s inequalities apparently implies that at least one of two beliefs, which are accepted as true in almost all scientific practice, must be false. These beliefs are: that effects propagating at infinite velocity do not exist, and that natural phenomena occur independently of being observed. Giving up any one of these two beliefs (usually known together as Local Realism) is controversial. For many years, many theories have been proposed to reconcile the violation of Bell’s inequalities with Local Realism, but none has been fully successful. In this paper, it is recalled that Bell’s inequalities are equivalent to the conditions to decide the completeness of a theory according to Boolean logic. Therefore, any theory aimed to violate Bell’s inequalities must start by giving up Boolean logic. Two problems are therefore defined: the “soft” one is to explain the violation of Bell’s inequalities without violating (non-Boolean) Local Realism; the “hard” one is to predict the time values when single particles are detected, in such a way that the resulting number of coincidences violates Bell’s inequalities. A simple hidden variables model is introduced, which solves the “soft” problem even in an ideal setup. This is possible thanks to the use of vectors in real space as the hidden variables and the corresponding operation (projection), which do not hold to Boolean logic. This model reconciles the violation of Bell’s inequalities with Local Realism and should end a controversy that has lasted for decades. Regarding the “hard” problem, the introduced model is as incomplete as Quantum Mechanics is. It is claimed that solving the “hard” problem requires devising a new kind of quantum computer, which should be able to accept (non-Boolean) hidden variable values as input data and to replace the statistical Born’s rule of Quantum Mechanics with a deterministic threshold condition.