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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Quantum Malus law for composite systems as a hidden-variable theory

Mladen Pavičić

Published Nov 15, 1990• 2 citations
Publisher

Abstract

By referring to only two positions of measuring devices, thus avoiding the counterfactual reasoning of the Bell result, a plausibly general proof is provided that the distribution function of the hidden variables can be neither a continuous nor a singular local function but must be at least a singular nonlocal one. The recognition of Scully and Milonni's theory as a theory which makes the quantum Malus law work for composite systems was the clue to the proof.

DOI
10.1103/physrevd.42.3594

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By referring to only two positions of measuring devices, thus avoiding the counterfactual reasoning of the Bell result, a plausibly general proof is provided that the distribution function of the hidden variables can be neither a continuous nor a singular local function but must be at least a singular nonlocal one. The recognition of Scully and Milonni's theory as a theory which makes the quantum Malus law work for composite systems was the clue to the proof.
The basic Leggett inequalities, i.e. those inequalities in which the particular assumptions of Leggett's hidden-variable model (e.g. Malus law) were not yet introduced, are usually derived using hidden-variable distributions of probabilities (although in some cases completely general, positive probabilities would lead to the same result). This fact creates sometimes the illusion that these basic inequalities are a belonging of the hidden-variable theories and are bound to contradict the quantum theory. In the present text the basic Leggett inequalities are derived in the most general way, i.e. no assumption is made that the distribution of probabilities would result from some wave function, or from some set of classical variables. The consequence is that as long as one and the same probability distribution is used in the calculus of all the averages appearing in the basic Leggett inequalities, no contradiction may occur. These inequalities may be violated only when different averages are calculated with different distributions, for example, some of them calculated with the quantum formalism and the others with some distribution of classical parameters.
A local hidden variables solution of Malus’s Law and a circumvention of Bell’s Theorem. The solution for Bell assumes that antiparticles actually do travel backwards in time and therefore a Bell experiment begins not at the usual source of particle pairs but at the measurement of positrons and ends at the measurement of their paired partner electrons. The solution for Malus’s Law assumes that the distribution of hidden variables in a polarised beam can be deduced by reverse-engineering Malus’s Law intensity calculations. Malus’s Law and Bell’s Experiment can therefore each be explained using local hidden variables. A computer program is given to provide results of a Stern Gerlach detector using local hidden variables in a particle-at-a-time simulation.
Bell's theorem proves the incompatibility between quantum mechanics and local realistic hidden-variable theories. In this paper we show that, contrary to a common belief, the theoretical proof of Bell's theorem is not affected by counterfactual reasoning. Then, we demonstrate that the experimental verification of this theorem may be affected in an unknowable way by our ignorance about the probability distribution of the hidden variables. Our study is based on the standard theory of random variables, and lays the groundwork for a critical rethinking of Bell's theorem and its consequences.
The main purpose of this paper is to provide a new proof of the impossibility of local theories of hidden variables based on simpler and more general assumptions than those used by Bell (1964, 1966), Wigner (1970), and Suppes and Zanotti (1976). In particular, no assumptions requiring specific quantum mechanical calculations are required. They are replaced by the principle of exchangeability and the principle of identical conditional distributions given the hidden variable. The results may be most easily discussed in terms of a system of two spin-½ particles initially in the singlet state, but generalizations to other quantum mechanical systems of a similar nature are apparent. In essential terms, the analysis of Bell, stated in a particularly clear form by Wigner, depends upon the following assumptions, which we state in intuitive form. (Mathematically explicit axioms are formulated in Appendix A.) 1. Axial symmetry . For any direction of the measuring apparatus the expected spin is 0, where spin is measured by +1 and -1 for spin +½ and spin -½, respectively. Further, the expected product of the spin measurements is the same for different orientations of the measuring apparatuses, as long as the angle between the measuring apparatuses remains the same. 2. Opposite measurement for same orientation . The correlation between the spin measurements is – 1 if the two measuring apparatuses have the same orientation. 3. Independence of λ . The expectation of any function of λ is independent of the orientation of the measuring apparatus.
Experiments motivated by Bell's theorem have led some physicists to conclude that quantum theory is nonlocal. However, the theoretical basis for such claims is usually taken to be Bell's Theorem, which shows only that if certain predictions of quantum theory are correct, and a strong hidden-variable assumption is valid, then a certain locality condition must fail. This locality condition expresses the idea that what an experimenter freely chooses to measure in one spacetime region can have no effect of any kind in a second region situated spacelike relative to the first. The experimental results conform closely to the predictions of quantum theory in such cases, but the most reasonable conclusion to draw is not that locality fails, but rather that the hidden-variable assumption is false. For this assumption conflicts with the quantum precept that unperformed experiments have no outcomes. The present paper deduces the failure of this locality condition directly from the precepts of quantum theory themselves, in a way that generates no inconsistency or any conflict with the predictions of relativistic quantum field theory.
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.