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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Some evaluations of Bell's inequality for particles of arbitrary spin

Mats Ögren

Published Apr 15, 1983• 13 citations
Publisher

Abstract

Bell's inequality is applied to a gedanken experiment, where a spin singlet decays into two spin-$s$ particles. This is done by defining dichotomic observables depending on the spin components. The range of settings, for which the inequality is violated by quantum mechanics, is calculated for certain values of $s$. The classical limit is taken and the results are compared to those obtained, by an alternative method, by Mermin.

DOI
10.1103/physrevd.27.1766

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Bell's inequality is applied to a gedanken experiment, where a spin singlet decays into two spin-sss particles. This is done by defining dichotomic observables depending on the spin components. The range of settings, for which the inequality is violated by quantum mechanics, is calculated for certain values of sss. The classical limit is taken and the results are compared to those obtained, by an alternative method, by Mermin.
The Mermin-Klyshko inequality for n spin-12\frac{1}{2}21​ particles and two dichotomic observables is generalized to n spin-sss particles and two maximal observables. It is shown that some multiparty multilevel Greenberger-Horne-Zeilinger states [A. Cabello, Phys. Rev. A 63, 022104 (2001)] maximally violate this inequality for any s.s.s. For a fixed n,n,n, the magnitude of the violation is constant for any s,s,s, which provides a simple demonstration and generalizes the conclusion reached by Gisin and Peres for two spin-sss particles in the singlet state [Phys. Lett. A 162, 15 (1992)]. For a fixed s,s,s, the violation grows exponentially with n,n,n, which provides a generalization to any s of Mermin's conclusion for n $\mathrm{spin}\ensuremath{-} \frac{1}{2}$ particles [Phys. Rev. Lett. 65, 1838 (1990)].
John Ju Sakurai's classical book in quantum mechanics makes a very illuminative presentation that studies entangled states in a two spin s=1/2 particles system in a singlet state. A Bell's inequality emerges as a consequence. Bell's inequality is a relationship among observables that discriminates between Einstein's locality principle and the nonlocal point of view of orthodox quantum mechanics. Following Sakurai's style we propose, by making natural induction, a generalization for Bell's inequality for any two spin-s particles in a singlet state (s integer or half-integer). This inequality is expressed as a function of a θ parameter, which is a measure of the angle between two possible directions in which the spin is measured. Besides the expression for this general inequality we have found that: (a) for any finite half-integer spin Bell's inequality is violated for some interval of the θ-parameter. The right limit of this interval is fixed and equal to π/2, while the left one comes closer and closer to this value as spin number grows. A function fit shows clearly that the size of this θ-interval over which Bell's inequality is violated diminishes asymptotically to zero as 1/s1/2; (b) an analogous behavior for any finite integer spin. For large spins the disagreement between Einstein's locality principle and the nonlocal point of view in orthodox quantum mechanics disappears.
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.
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​.
We formulate the expectation value of the Bell-Żukowski operator acting on qubit states of a two-particle Bell experiment. By using the equivalence between a set of N copies of a two-qubit experiment and a standard two-setting Bell experiment in an entangled 2N-particle state, we obtain an inequality, which we may call the Bell-Żukowski inequality. It determines whether the measured correlation functions of two-particle states can be modeled locally and realistically. In this Bell experiment of two particles, the conflict between local realism and quantum mechanics is discussed in conjunction with the violation of the Bell-Żukowski inequality. The main point of the result is that the Bell-Żukowski operator can be represented by the Bell-Mermin operator. PACS numbers: 03.65.Ud, 03.67.Mn
We present a macroscopical mechanistic classical laboratory situation, and a classical macroscopical entity, and coincidence measurements on this entity, that lead to a violation of the Bell inequalities corresponding to these coincidence measurements. The violation that we obtain with these coincidence measurements is exactly the same as the violation of the Bell inequalities by the well known coincidence measurements of the quantum entity of two spin 1/2 particles in a singlet spin state. With this we mean that it gives rise to the same numerical values for the expectation values and the same numerical value 2•√2 for the expression used in the Bell inequalities.