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Physical Review A

Violation of local realism by a system withNspin- 12 particles

Xiao-Hua Wu, Hong-Shi Zong

Published Sep 19, 2003• 17 citations
Publisher

Abstract

Recently, it was found that Mermin's inequalities may not always be optimal for the refutation of a local realistic description [Phys. Rev. Lett. 88, 210402 (2002)]. To complete this work, we derive an inequality for the Greenberger-Horne-Zeilinger-type pure state for a system with N spin-$\frac{1}{2}$ particles and the violation of the inequality can be shown for all the non product pure states. Mermin's inequality for a system of N spin-$\frac{1}{2}$ particles and Gisin's theorem for a system of two spin-$\frac{1}{2}$ particles are both included in our inequality.

DOI
10.1103/physreva.68.032102

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Recently, it was found that Mermin's inequalities may not always be optimal for the refutation of a local realistic description [Phys. Rev. Lett. 88, 210402 (2002)]. To complete this work, we derive an inequality for the Greenberger-Horne-Zeilinger-type pure state for a system with N spin-12\frac{1}{2}21​ particles and the violation of the inequality can be shown for all the non product pure states. Mermin's inequality for a system of N spin-12\frac{1}{2}21​ particles and Gisin's theorem for a system of two spin-12\frac{1}{2}21​ particles are both included in our inequality.
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)].
It is proved that all states of three spin-12\frac{1}{2}21​ particles exhibiting an ``all versus nothing'' contradiction between quantum mechanics and the local realism of Einstein, Podolsky, and Rosen are exactly the Greenberger-Horne-Zeilinger (GHZ) states and the states obtained from them by local unitary transformations. The proof is obtained by showing that there are at most four elements (except for a different sign) in a set of mutually commuting nonlocal spin observables in the three-qubit system and using the certain algebraic properties that Pauli's matrices satisfy. We show that only does such a set of four nonlocal spin observables present a Greenberger-Horne-Zeilinger-Mermin-like argument. This also reveals the equivalence between the GHZ theorem and maximal violation of the Bell inequality.
We investigate the violation factor of the original Bell-Mermin inequality. Until now, we have used an assumption that the results of measurement are . In this case, the maximum violation factor is as follows: and . The quantum predictions by n-partite Greenberger-Horne-Zeilinger state violate the Bell-Mermin inequality by an amount that grows exponentially with n. Recently, a new measurement theory is proposed [K. Nagata and T. Nakamura, International Journal of Theoretical Physics, 49, 162 (2010)]. The values of measurement outcome are . Here we use the new measurement theory. We consider a multipartite GHZ state. We use the original Bell-Mermin inequality. It turns out that the original Bell-Mermin inequality is satisfied irrespective of the number of particles. In this case, the maximum violation factor is as follows: and . Thus the original Bell-Mermin inequality is satisfied by the new measurement theory. We propose the following conjecture: All the two-orthogonal-settings experimental correlation functions admit local realistic theories irrespective of a state if we use the new measurement theory.
Gisin's theorem states that any pure entangled state of two particles violates Clauser-Horne-Shimony-Holt inequality which is a bound on correlations between measurements on two particles obtained within local realistic picture. We find that this theorem cannot be generalized to all multi-particle states for the case where measurements on each particle can be chosen between two arbitrary dichotomic observables. In particular we find a family of pure entangled states of N qubits which do not violate any Bell inequality for correlations between such measurements. We also find that Mermin-Ardehali-Belinskii-Klyshko inequalities may not always be optimal for refutation of local realistic description of correlations.