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AIP conference proceedings

A tutorial on quantum distribution functions for spin-12 systems and Einstein-Podolsky-Rosen correlations

Marlan O. Scully, Hwang Lee, Eduardo García Gómez, Roberto Ortega-Martı́nez

Published Jan 1, 1999• 0 citations
Publisher

Abstract

In earlier paper entitled How to make quantum mechanics look like a hidden-variable theory and vice versa, Phys. Rev. D 28, 2477 (1983) [1], we developed a quantum distribution function analysis for spin-12 systems, which had many of the characteristics of a hidden variable theory but was rigorously quantum mechanical. In the same paper, we also presented a hidden variable theory which was modeled after the Wigner like quantum distribution analysis and which agreed with the predictions of Einstein-Podolsky-Rosen (EPR) experiments but was non-local. These considerations are useful in understanding various subtle aspects of locality and reality in EPR type experiments. This has not been widely appreciated by students of philosophy and of the foundations of quantum mechanics. This is at least in part no doubt because the mathematical underpinnings are of a somewhat different nature than those usually associated with quantum mechanics. That is, the elements of quantum distribution theory are not the usual material of quantum mechanics course. The main goal of the present notes is therefore to provide a tutorial on quantum distribution theory as applied to spin one half particles. To that end, the original Phys. Rev. paper is included, as are our extensions of that paper. The technical variations, which are naturally included in an abbreviated form in the Phys. Rev. paper, are here presented as a series of worked problems.

DOI
10.1063/1.58230

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In earlier paper entitled How to make quantum mechanics look like a hidden-variable theory and vice versa, Phys. Rev. D 28, 2477 (1983) [1], we developed a quantum distribution function analysis for spin-12 systems, which had many of the characteristics of a hidden variable theory but was rigorously quantum mechanical. In the same paper, we also presented a hidden variable theory which was modeled after the Wigner like quantum distribution analysis and which agreed with the predictions of Einstein-Podolsky-Rosen (EPR) experiments but was non-local. These considerations are useful in understanding various subtle aspects of locality and reality in EPR type experiments. This has not been widely appreciated by students of philosophy and of the foundations of quantum mechanics. This is at least in part no doubt because the mathematical underpinnings are of a somewhat different nature than those usually associated with quantum mechanics. That is, the elements of quantum distribution theory are not the usual material of quantum mechanics course. The main goal of the present notes is therefore to provide a tutorial on quantum distribution theory as applied to spin one half particles. To that end, the original Phys. Rev. paper is included, as are our extensions of that paper. The technical variations, which are naturally included in an abbreviated form in the Phys. Rev. paper, are here presented as a series of worked problems.
The quantum theory of a singlet spin-\textonehalf{} system is developed in terms of angular variables using a quantum-distribution-function technique. These calculations demonstrate a much closer correspondence between quantum mechanics and certain hidden-variable theories than was previously appreciated. It is found that a new type of hidden-variable theory is suggested by the quantum-distribution-function treatment of the Einstein-Podolsky-Rosen-Bohm spin-spin correlation problem which is in agreement with quantum theory but is "nonlocal."
Introduction The paradox of Einstein, Podolsky and Rosen was advanced as an argument that quantum mechanics could not be a complete theory but should be supplemented by additional variables. These additional variables were to restore to the theory causality and locality. In this note that idea will be formulated mathematically and shown to be incompatible with the statistical predictions of quantum mechanics. It is the requirement of locality, or more precisely that the result of a measurement on one system be unaffected by operations on a distant system with which it has interacted in the past, that creates the essential difficulty. There have been attempts to show that even without such a separability or locality requirement no ‘hidden variable’ interpretation of quantum mechanics is possible. These attempts have been examined elsewhere and found wanting. Moreover, a hidden variable interpretation of elementary quantum theory has been explicitly constructed. That particular interpretation has indeed a grossly non-local structure. This is characteristic, according to the result to be proved here, of any such theory which reproduces exactly the quantum mechanical predictions. Formulation With the example advocated by Bohm and Aharonov, the EPR argument is the following. Consider a pair of spin one-half particles formed somehow in the singlet spin state and moving freely in opposite directions. Measurements can be made, say by Stern–Gerlach magnets, on selected components of the spins σ 1 and σ 2 .
We examine the properties and behavior of the original Einstein-Podolsky-Rosen (EPR) wave function [Phys. Rev. 47, 777 (1935)] and related Gaussian-correlated wave functions. We assess the degree of entanglement of these wave functions and consider an argument of Bell [Ann. (N.Y.) Acad. Sci. 480, 263 (1986)] based on the Wigner phase-space distribution [Phys. Rev. 40, 749 (1932)], which implies that the original EPR correlations can accommodate a local hidden-variable description. We extend Bell's analysis to the related Gaussian wave functions. We then show that it is possible to identify definite nonlocal aspects for the original EPR state and related states. We describe possible experiments that would demonstrate these nonlocal features through violations of Bell inequalities. The implications of our results, and in particular their relevance for the causal interpretation of quantum mechanics, are considered.
A critical reconsideration of the EPR (Einstein-Podolsky-Rosen) paper shows that the EPR argument can be developed without using the concept of `element of physical reality', thus eliminating any philosophical element in the logical chains of the paper. Deprived of its philosophical ornament, the EPR argument plainly reduces to require what quantum mechanics can not do: to assign definite values to two incompatible physical quantities. Hidden variables theories built up according to Bell - type theorems are formulated on the basis of the assumption that the locality condition implies the statistical independence between two measurements space - like separated. This assumption is valid only with the additional one that statistical dependence between two measurements requires a causal connection between them. This additional assumption rules out the possibility that statistical dependence may due to an intrinsic property of the physical system under study. Therefore, hidden variables theories are built up with a restriction which leads them to be disproved by experiment. Quantum mechanical non - locality, invoked for describing EPR - type experiments, is strictly connected to the hypothesis (NDV hypothesis) according to which the twin photons of entangled pairs do not have a definite polarization before measurements. Both hypotheses are used only for describing EPR experiments and not for making predictions. Therefore, they can be dropped without reducing the predictive power of quantum mechanics concerning entangled photons pairs. Furthermore, both hypotheses can be experimentally tested by a modification of a standard experimental apparatus designed for studying entangled photons pairs.
1 History of the Einstein-Podolsky-Rosen Paradox.- 2 Are Faster-Than-Light Influences Necessary?.- 3 All the Inequalities of Einstein Locality.- 4 Einstein-Podolsky-Rosen Experiments Using the Decays of ?c or J/? into ??? ? ?-?p?+p?.- Chapter5 Einstein-Podolsky-Rosen Paradox for the K0-K?0 and B0-B?0 Systems.- 6 Even Local Probabilities Lead to the Paradox.- 7 The Experimental Investigation of the Einstein-Podolsky-Rosen Question and Bell's Inequality.- 8 Rapisarda's Experiment: Testing Quantum Mechanics versus Local Hidden-Variable Theories with Dichotomic Analyzers.- 9 Nonlocality and the Einstein-Podolsky-Rosen Experiment as Understood through the Quantum-Potential Approach.- 10 Interpretation of the Einstein-Podolsky-Rosen Effect in Terms of a Generalized Causality.- 11 Quantum Action-at-a-Distance: The Mystery of Einstein-Podolsky-Rosen Correlations.- 12 Particle Trajectories and Quantum Correlations.- 13 Bell's Inequality and the Nonergodic Interpretation of Quantum Mechanics.- 14 An Extended-Probability Response to the Einstein-Podolsky-Rosen Argument.- 15 The Search for Hidden Variables in Quantum Mechanics.- 16 Variable Detection Probability Models for Einstein-Podolsky-Rosen-Type Experiments.- 17 Stochastic Electrodynamics and the Einstein-Podolsky-Rosen Argument.- 18 Explicit Calculations with a Hidden-Variable Spin Model.- 19 Symmetric and Asymmetric Models for Atomic Cascade Experiments.
Bohm's spin polarization modification of the paradox of Einstein, Podolsky, and Rosen has served as the basis for numerous analyses of the roles of hidden variables and Einstein locality in quantum mechanics. The present note points out how corresponding analyses can be carried out on the original formulation of Einstein, Podolsky, and Rosen (EPR) in terms of observations of position and momentum. Concrete conceptual experiments are proposed to furnish tangible illustrations of EPR tests of reality, and a position-momentum variant of Bell's inequality is formulated. Implications regarding joint measurements of position and momentum, delayed choice experiments, and superluminal communication are discussed briefly.