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TMF, 2004 Volume 140, Number 2, Pages 284–296 (Mi tmf87)

This article is cited in 17 papers

Bell's Inequality for Two-Particle Mixed Spin States

V. A. Andreev, V. I. Man'ko

P. N. Lebedev Physical Institute, Russian Academy of Sciences

Abstract: We derive the Bell–Clauser–Horne–Shimony–Holt inequalities for two-particle mixed spin states both in the conventional quantum mechanics and in the hidden-variables theory. We consider two cases for the vectors $\vec a$, $\vec b$, $\vec c$, $\vec d$ specifying the axes onto which the particle spins of a correlated pair are projected. In the first case, all four vectors lie in the same plane, and in the second case, they are oriented arbitrarily. We compare the obtained inequalities and show that the difference between the predictions of the two theories is less for mixed states than for pure states. We find that the inequalities obtained in quantum mechanics and the hidden-variables theory coincide for some special states, in particular, for the mixed states formed by pure factorable states. We discuss the points of similarity and difference between the uncertainty relations and Bell's inequalities. We list all the states for which the right-hand side of the Bell–Clauser–Horne–Shimony–Holt inequality is identically equal to zero.

Keywords: quantum mechanics, hidden variables, Bell inequality, mixed states, spin states, quantum tomography.

Received: 03.09.2003

DOI: 10.4213/tmf87


 English version:
Theoretical and Mathematical Physics, 2004, 140:2, 1135–1145

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