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European Journal of Physics

Total quantum state in the Einstein–Podolsky–Rosen–Bohm experiment with identical particles

Sebastiano Sonego

Published Mar 21, 2019• 0 citations
Publisher

Abstract

Abstract The spin state in the Einstein–Podolsky–Rosen–Bohm gedankenexperiment with identical particles is supplemented by the spatial part. This allows one to extract all the information needed in a typical EPR argument, without requiring semi-intuitive steps. Local spin operators are introduced, to describe measurements of spin in given regions of space.

DOI
10.1088/1361-6404/ab1201

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Abstract The spin state in the Einstein–Podolsky–Rosen–Bohm gedankenexperiment with identical particles is supplemented by the spatial part. This allows one to extract all the information needed in a typical EPR argument, without requiring semi-intuitive steps. Local spin operators are introduced, to describe measurements of spin in given regions of space.
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 .
The status of our experimental realization of Bohm’s spin-1/2 particle version of the Einstein-Podolsky-Rosen gedankenexperiment is discussed. This new experiment provides a definitive test of the, as yet, untested strong Bell inequalities. It closes the two widely recognized loopholes associated with previous experiments, and has the interesting new feature that the entangled state exists for milliseconds rather than nanoseconds. The spin one-half particles are two 199 Hg atoms in an entangled nuclear spin state with total nuclear spin zero. Measurement of the nuclear spin correlation is achieved by detection of each atom using a spin state selective two photon excitation-ionization scheme.
In this paper we calculate with full details Einstein-Podolsky-Rosen spin correlations in the framework of nonrelativistic quantum mechanics. We consider the following situation: two-particle state is prepared (we consider separately distinguishable and identical particles and take into account the space part of the wave function) and two observers in relative motion measure the spin component of the particle along given directions. The measurements are performed in bounded regions of space (detectors), not necessarily simultaneously. The resulting correlation function depends not only on the directions of spin measurements but also on the relative velocity of the observers.
Prediction of the correlation between the data does not cause any problems in an ideal Einstein–Podolsky–Rosen (EPR) experiment, but if both measurements are not really performed at the same time one should take into account that a particle is irreversibly absorbed by a detector during the measurement. We calculate with full details the destruction of quantum states during the spin measurement of a particle involving localization of a particle in a detector in a system of two distinguishable particles of spin-½ and consider an example of the destruction in the configuration typical for EPR experiments.
Bohm version of the EPR paradox Einstein, Podolsky, and Rosen (EPR) were concerned with the following issue. Given two spatially separated quantum systems A and B and an appropriate initial entangled state, a measurement of a property on system A can be an indirect measurement of B in the sense that from the outcome of the A measurement one can infer with probability 1 a property of B , because the two systems are correlated. There are cases in which either of two properties of B represented by noncommuting projectors can be measured indirectly in this manner, and EPR argued that this implied that system B could possess two incompatible properties at the same time, contrary to the principles of quantum theory. In order to understand this argument, it is best to apply it to a specific model system, and we shall do so using Bohm's formulation of the EPR paradox in which the systems A and B are two spin-half particles a and b in two different regions of space, with their spin degrees of freedom initially in a spin singlet state (23.2). As an aid to later discussion, we write the argument in the form of a set of numbered assertions leading to a paradox: a result which seems plausible, but contradicts the basic principles of quantum theory. The assertions E1–E4 are not intended to be exact counterparts of statements in the original EPR paper, even when the latter are translated into the language of spin-half particles. However, the general idea is very similar, and the basic conundrum is the same.
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.