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Physical Review Letters

Bell Inequalities with a Range of Violation that Does Not Diminish as the Spin Becomes Arbitrarily Large

Anupam Garg, N. David Mermin

Published Sep 27, 1982• 92 citations
Publisher

Abstract

Some new, powerful, and very simple Bell inequalities are derived. They establish that for two spin-$s$ particles in the singlet state, the quantum-theoretic Einstein-Podolsky-Rosen correlations for spin measurements along all pairs of directions from a set of $N$ axes are incompatible with local realism for any set of distinct coplanar axes when $N=3$, and for any set of distinct axes whatever when $N=4$, 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.

DOI
10.1103/physrevlett.49.901

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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.
Some simple inequalities which demonstrate the incompatibility of local realism with quantum theory are derived. They establish, for the first time, necessary conditions for violation of the generalized spin-sss Bell inequalities for a set of three distinct noncoplanar axes. For s=12s=\frac{1}{2}s=21​, however, these inequalities are equivalent to Wigner's results, thus giving necessary and sufficient conditions.
We review the computation of correlations of successive projections of the spin onto axes for spin-1/2 particles and EPRB pairs (in the Singlet state). We assume forms of Realism (at least as general as the Predictive Hidden Variables in the classical Bell's Theory), to review one particle correlations before we further assume Locality and analyze EPRB pairs. To our surprise, we find that the Abstract Bell Inequalities with three or four axes cannot have physical meaning i.e., two spin projections cannot belong to one particle or some correlations that one would need cannot be evaluated. A recent version of Bell's Theory without Locality turns out to be unaffected by our results that refocus the open issues on "realism" rather than "local realism", inviting us to leave locality alone so to speak.
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The singlet state of two spin-32\frac{3}{2}23​ particles allows a proof of Bell's theorem without inequalities with two distinguishing features: any local observable can be regarded as an Einstein-Podolsky-Rosen element of reality, and the contradiction with local realism occurs not only for some specific local observables but for any rotation whereof.
The quantitative quantum-mechanical analysis of the Einstein-Podolsky-Rosen experiment for correlated particles of arbitrary spin sss is shown to contradict a generalized form of Bell's inequality, for suitable orientations of the detectors. As the classical ($s\ensuremath{\rightarrow}\ensuremath{\infty}$) limit is approached, the range of angles for which the contradiction arises vanishes as 1s\frac{1}{s}s1​.
Two aspects of the Clauser-Horne conditions for the compatibility with local realism of measured spin-\textonehalf{} Einstein-Podolsky-Rosen correlations are investigated in the spin-sss case. (1) A new set of necessary conditions is given for compatibility with local realism. These conditions are violated for a large range of geometries. The range does not diminish with increasing sss, if the observed correlations are sufficiently near to the quantum-theoretic predictions. (2) A simple counterexample is given to the spin-1 generalization of a recent conjecture that the conditions tested by the Clauser-Horne spin-\textonehalf{} inequalities are sufficient as well as necessary for compatibility of the data with local realism.