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Reviews of Modern Physics

Pairs of two-level systems

U. Fano

Published Oct 1, 1983• 195 citations
Publisher

Abstract

Correlations between the orientations of two particles with spin \textonehalf{} are formulated in terms of a complete set of operators appropriate to display the spectrum of oscillations under the influence of the spin interactions. The operators are classified for this purpose according to space and permutation symmetries. The treatment is intended as a model for any pair of two-level systems. Its extension to multilevel systems and the eventual emergence of dissipative behavior are outlined. The correlation of two-photon polarizations in the Einstein-Rosen-Podolsky paradox is described as an illustration.

DOI
10.1103/revmodphys.55.855

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Correlations between the orientations of two particles with spin \textonehalf{} are formulated in terms of a complete set of operators appropriate to display the spectrum of oscillations under the influence of the spin interactions. The operators are classified for this purpose according to space and permutation symmetries. The treatment is intended as a model for any pair of two-level systems. Its extension to multilevel systems and the eventual emergence of dissipative behavior are outlined. The correlation of two-photon polarizations in the Einstein-Rosen-Podolsky paradox is described as an illustration.
A local realistic model for quantum mechanics of two-particle Einstein-Podolsky-Rosen pairs is proposed. In this model, it is the strict obedience of conservation laws in each event at the quantum level that uphold the perfect correlation of two spatially-separated particles, instead of nonlocality in the orthodox formulation of quantum mechanics. Therefore, one can conclude that all components of the spin of two particles, and the position and momentum of a particle can be measured simultaneously. The proposed model yields the same statistical prediction on an ensemble of individual particles as the orthodox formulation does. This suggests that the wave function is not a complete description of individual particle as assumed in the orthodox formulation, but only a statistical description of an ensemble of particles.
We consider the Einstein-Podolsky-Rosen correlation between two spins with arbitrary j values using the phase-space description of the angular-momentum operators [G. S. Agarwal, Phys. Rev. A 24, 2889 (1981)]. The phase-space formulation provides insight into hidden-variable theories and enables us to understand why Bell's inequalities are violated. We use the analogs of the Wigner function and the P function for the analysis of the correlations. We show that the Wigner function for the singlet state becomes positive in the limit j\ensuremath{\rightarrow}\ensuremath{\infty}. We also show the extent to which Bell's inequalities can be violated for higher spins and present physical quantities, the correlations between which would violate Bell's inequalities.
The formula for the correlation function of spin measurements of two particles in two moving inertial frames is derived within Lorentz-covariant quantum mechanics formulated in the absolute synchronization framework. These results are the first exact Einstein-Podolsky-Rosen correlation functions obtained for Lorentz-covariant quantum-mechanical systems in moving frames under physically acceptable conditions, i.e., taking into account the localization of the particles during the detection and using the spin operator with proper transformation properties under the action of the Lorentz group. Some special cases and approximations of the calculated correlation function are given. The resulting correlation function can be used as a basis for a proposal of a decisive experiment for a possible existence of a quantum-mechanical preferred frame.