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Physics Letters A

The classification of two-particle spin states and generalized Bell inequalities

V. A. Andreev, Vladimir I. Man’ko

Published Apr 1, 2001• 17 citations
Publisher
DOI
10.1016/s0375-9601(01)00130-x

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It is generally believed that Bell's inequality holds for the case of entangled states, including two correlated particles or special states of a single particle. Here, we derive a single-particle Bell's inequality for two correlated spin states at two successive times, appealing to the statistical independence condition in an ideal experiment, for a locally causal hidden variables theory. We show that regardless of the locality assumption, the inequality can be violated by some quantum predictions.
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.