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Physical Review A

Perspective of Einstein-Podolsky-Rosen spin correlations in the phase-space formulation for arbitrary values of the spin

G. S. Agarwal

Published Jun 1, 1993• 39 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.47.4608

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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 spin-jjj extension of Bohm's version of the Einstein-Podolsky-Rosen experiment is is analysed in terms of the Wigner function when the two spins are in a singlet state. This function is calculated for all jjj, and it is shown that just as Bell inequalities are violated with undiminished range and magnitude for arbitarirly large jjj, this function does not become less negative. On the contrary, the oscillations between positive and negative values grow both in frequency and amplitude. It is argued that this is an alternative way to grasp the approach to classical behavior with increasing quantum number.
Generalizations of Bell's inequalities for Einstein Podolsky and Rosen entangled correlations for arbitrary spin s are discussed.Violations of local realism and the classical limit of quantum nonlocality for large s are discussed.The role of the polarization and of the spin alignment in the violation of the Bell inequalities are investigated.A Bayes analysis of entangled correlations is performed using the nonlocal quantum distribution.The local but nonpositive Einstein Podolsky and Rosen quantum distribution is investigated in the limit when s → .The classical limit of the quantum Malus 1aw for arbitrary spin s is formulated in terms of the spin patl integrals.
I derive the correlation function for a general theory of two-valued spin variables that satisfy the fundamental conservation law of angular momentum. The unique theory-independent correlation function is identical to the quantum mechanical correlation function. I prove that any theory of correlations of such discrete variables satisfying the fundamental conservation law of angular momentum violates the Bell’s inequalities. Taken together with the Bell’s theorem, this result has far reaching implications. No theory satisfying Einstein locality, reality in the EPR-Bell sense, and the validity of the conservation law can be constructed. Therefore, all local hidden variable theories are incompatible with fundamental symmetries and conservation laws. Bell’s inequalities can be obeyed only by violating a conservation law. The implications for experiments on Bell’s inequalities are obvious. The result provides new insight regarding entanglement, and its measures. PACS Numbers: 03.65.Ta, 03.65.Ud The main result of this paper is the proof that a general physical theory of correlations of two-valued spin projections violate the Bell’s inequalities if the theory satisfies the conservation law for angular momentum on the average. The proof is applicable to local hidden variable theories, as well as
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.
We calculate the joint probabilities and the correlation function in Einstein-Podolsky-Rosen--type experiments with a massive vector boson in the framework of quantum field theory. We report on the strange behavior of the correlation function (and the probabilities)---the correlation function, which in the relativistic case still depends on the particle momenta, for some fixed configurations has local extrema. We also show that relativistic spin-1 particles violate some Bell inequalities more than nonrelativistic ones and that the degree of violation of the Bell inequality is momentum dependent.
Some new, powerful, and very simple Bell inequalities are derived. They establish that for two spin-sss particles in the singlet state, the quantum-theoretic Einstein-Podolsky-Rosen correlations for spin measurements along all pairs of directions from a set of NNN axes are incompatible with local realism for any set of distinct coplanar axes when N=3N=3N=3, and for any set of distinct axes whatever when N=4N=4N=4, with the possible exception of sets restricted by one of the constraints $\stackrel{^}{a}\ifmmode\pm\else\textpm\fi{}\stackrel{^}{b}\ifmmode\pm\else\textpm\fi{}\stackrel{^}{c}\ifmmode\pm\else\textpm\fi{}\stackrel{^}{d}=0$. These results hold for any value of the spin.