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arXiv:quant-ph/0211075 · Physical Review Letters

All-Versus-Nothing Violation of Local Realism for Two Entangled Photons

Zeng‐Bing Chen, Jian-Wei Pan, Yong-De Zhang, Časlav Brukner, Anton Zeilinger

Published Apr 24, 2003• 77 citations
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Abstract

It is shown that the Greenberger-Horne-Zeilinger theorem can be generalized to the case with only two entangled particles. The reasoning makes use of two photons which are maximally entangled both in polarization and in spatial degrees of freedom. In contrast to Cabello's argument of "all versus nothing" nonlocality with four photons [Phys. Rev. Lett. 87, 010403 (2001)]], our proposal to test the theorem can be implemented with linear optics and thus is well within the reach of current experimental technology.

DOI
10.1103/physrevlett.90.160408

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It is shown that the Greenberger-Horne-Zeilinger theorem can be generalized to the case with only two entangled particles. The reasoning makes use of two photons which are maximally entangled both in polarization and in spatial degrees of freedom. In contrast to Cabello's argument of "all versus nothing" nonlocality with four photons [Phys. Rev. Lett. 87, 010403 (2001)]], our proposal to test the theorem can be implemented with linear optics and thus is well within the reach of current experimental technology.
It is shown that the Greenberger-Horne-Zeilinger theorem can be generalized to the case with only two entangled particles. The reasoning makes use of two photons which are maximally entangled both in polarization and in spatial degrees of freedom. In contrast to Cabello's argument of "All versus Nothing" nonlocality with four photons [Phys. Rev. Lett. 87, 010403 (2001)], our proposal to test the theorem can be implemented with linear optics and thus is well within the reach of current experimental technology.
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.
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A recent proof of Bell's theorem without inequalities [A. Cabello, Phys. Rev. Lett. 86, 1911 (2001)] is formulated as a Greenberger-Horne-Zeilinger-type proof involving just two observers. On one hand, this new approach allows us to derive an experimentally testable Bell inequality which is violated by quantum mechanics. On the other hand, it leads to a new state-independent proof of the Kochen-Specker theorem and provides a wider perspective on the relations between the major proofs of no hidden variables.
The Greenberger-Horne-Zeilinger (GHZ) theorem exhibits an ``all versus nothing'' contradiction between Ernstein, Podelsky, and Rosen's local realism and quantum mechanics by using a maximally entangled state of three qubits. However, it is shown in this paper that no extension of GHZ's proof is possible for any entangled state of two qubits, including Bell states. The proof is obtained by showing the fact that there are no more than three distinct elements (except for a different sign) in a set of mutually commuting nonlocal spin observables of two qubits. This also leads to a generic setup of the Kochen-Specker theorem for the two-qubit system.