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arXiv:quant-ph/0312046

Comment on "Quantum nonlocality for a three-particle nonmaximally entangled state without inequalities"

Jose Luis Cereceda

Published Dec 5, 2003• 0 citations
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Abstract

In a recent Brief Report, Zheng [S-B. Zheng, PRA 66, 014103 (2002)] has given a proof of nonlocality without using inequalities for three spin-1/2 particles in the nonmaximally entangled state |psi> = cosθ|+++> + i sinθ|-->. Here we show that Zheng's proof is not correct. Indeed it is the case that, for the experiment considered by Zheng, the only state that admits a nonlocality proof without inequalities is the maximally entangled state.

DOI
10.48550/arxiv.quant-ph/0312046

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sComment on 'Local-realism violations in two-particle interferometry'

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In a recent Brief Report, Zheng [S-B. Zheng, PRA 66, 014103 (2002)] has given a proof of nonlocality without using inequalities for three spin-1/2 particles in the nonmaximally entangled state |psi> = cosθ|+++> + i sinθ|-->. Here we show that Zheng's proof is not correct. Indeed it is the case that, for the experiment considered by Zheng, the only state that admits a nonlocality proof without inequalities is the maximally entangled state.
The ladder proof of nonlocality without inequalities for two spin-12\frac{1}{2}21​ particles proposed by Hardy [in New Developments on Fundamental Problems in Quantum Physics, edited by M. Ferrero and A. van der Merwe (Kluwer, Dordrecht, 1997)] and Hardy and co-workers [Phys. Rev. Lett. 79, 2755 (1997)] works only for nonmaximally entangled states and goes through for 50% of pairs at the most. A similar ladder proof for two spin-111 particles in a maximally entangled state is presented. In its simplest form, the proof goes through for 17% of pairs. An extended version works for 100% of pairs. The proof can be extended to any maximally entangled state of two spin-sss particles (with s> 1)s>~1)s> 1).
In a recent article [Phys. Rev. A 53, R1927 (1996)] Wu et al. give a proof of nonlocality for the maximally entangled state of two particles without using inequalities. Here we point out that in order for their argument to go through it is necessary to consider a minimum total of six dimensions in Hilbert space. Indeed, the above argument is found to fail whenever the dimensionality of the product Hilbert space is reduced to four. The explanation for this failure is provided by the fact according to which no direct contradiction between the quantum-mechanical and locally realistic predictions can arise for any fraction of an ensemble of 2\ifmmode\times\else\texttimes\fi{}2 systems configured in the maximally entangled state.
The notions of three-particle entanglement and three-particle nonlocality are discussed in the light of Svetlichny's inequality [Phys. Rev. D 35, 3066 (1987)]. It is shown that there exist sets of measurements, which can be used to prove three-particle entanglement, but which are nevertheless useless at proving three-particle nonlocality. In particular, it is shown that the quantum predictions giving a maximal violation of Mermin's three-particle Bell inequality [Phys. Rev. Lett. 65, 1838 (1990)] can be reproduced by a hybrid hidden variables model in which nonlocal correlations are present only between two of the particles. It should be possible, however, to test the existence of both three-particle entanglement and three-particle nonlocality for any given quantum state via Svetlichny's inequality.
Quantum entanglement and quantum nonlocality of N -photon entangled states |ψ Nm ⟩ = C m [cos γ| N − m ⟩ 1 | m ⟩ 2 +e iθ m sin γ| m ⟩ 1 | N − m ⟩ 2 ] and their superpositions are studied. We point out that the relative phase θ m affects the quantum nonlocality but not the quantum entanglement for the state |ψ Nm ⟩. We show that quantum nonlocality can be controlled and manipulated by adjusting the state parameters of |ψ Nm ⟩, superposition coefficients and the azimuthal angles of the Bell operator. We also show that the violation of the Bell inequality can reach its maximal value under certain conditions. It is found that quantum superpositions based on |ψ Nm ⟩ can increase the amount of entanglement and give more ways to reach the maximal violation of the Bell inequality.