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arXiv:quant-ph/0202126 · Physical Review A

Bell’s inequality fornspin- s particles

Adán Cabello

Published Jun 5, 2002• 27 citations
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Abstract

The Mermin-Klyshko inequality for n spin-$\frac{1}{2}$ particles and two dichotomic observables is generalized to n spin-$s$ 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 $s.$ For a fixed $n,$ the magnitude of the violation is constant for any $s,$ which provides a simple demonstration and generalizes the conclusion reached by Gisin and Peres for two spin-$s$ particles in the singlet state [Phys. Lett. A 162, 15 (1992)]. For a fixed $s,$ the violation grows exponentially with $n,$ 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)].

DOI
10.1103/physreva.65.062105

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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 spin\ensuremath−12\mathrm{spin}\ensuremath{-} \frac{1}{2}spin\ensuremath−21​ particles [Phys. Rev. Lett. 65, 1838 (1990)].
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.
This Letter presents the first rigorous proof of the conjecture raised by Gisin and Bechmann-Pasquinucci [Phys. Lett. A 246, 1 (1998)]], that the Greenberger-Horne-Zeilinger states of n qubits and the states obtained from them by local unitary transformations are the unique states that maximally violate the Bell-Klyshko inequalities. The proof is obtained by using the certain algebraic properties that Pauli's matrices satisfy and some subtle mathematical techniques. Since all states obtained by local unitary transformations of a maximally entangled state are equally valid entangled states, we thus give a characterization of maximally entangled states of n qubits in terms of the Bell-type inequality.
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.
It was shown by Dur [Phys. Rev. Lett. 87,87,87, 230402 (2001)] that N (N> 4)(N>~4)(N> 4) qubits described by a certain one-parameter family F\mathcal{F}F of bound entangled states violate the Mermin-Klyshko inequality for N> 8.N>~8.N> 8. In this paper we prove that the states from the family F\mathcal{F}F violate Bell inequalities derived by \ifmmode \dot{Z}\else \.{Z}\fi{}ukowski and Kaszlikowski [Phys. Rev. A 56,56,56, R1682 (1997)], in which each observer measures three noncommuting sets of orthogonal projectors, for N> 7.N>~7.N> 7. We also derive a simple one-parameter family of entanglement witnesses that detect entanglement for all the states belonging to F.\mathcal{F}.F. It is possible that these entanglement witnesses could be generated by some Bell inequalities.