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Physical Review A

Asymmetric multipartite Greenberger-Horne-Zeilinger states and Bell inequalities

Darwin Gosal, Dagomir Kaszlikowski, L. C. Kwek, Marek Żukowski, C. H. Oh

Published Oct 29, 2004• 2 citations
Publisher

Abstract

We study the multiparticle generalized GHZ states. It has been shown that for an odd number of qubits and for a specific range of parameters, they do not violate any Bell inequality for correlation functions. We show here both analytically and numerically that, nevertheless, such states violate local realism, once a more detailed analysis of the correlations is made than the one allowed by correlation functions. The results imply that multiparticle Clauser-Horne-type inequalities involving probabilities are stronger tools for analyzing violations of local realism in multiparticle systems than inequalities involving the correlation functions.

DOI
10.1103/physreva.70.042106

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We study the multiparticle generalized GHZ states. It has been shown that for an odd number of qubits and for a specific range of parameters, they do not violate any Bell inequality for correlation functions. We show here both analytically and numerically that, nevertheless, such states violate local realism, once a more detailed analysis of the correlations is made than the one allowed by correlation functions. The results imply that multiparticle Clauser-Horne-type inequalities involving probabilities are stronger tools for analyzing violations of local realism in multiparticle systems than inequalities involving the correlation functions.
A family of Bell-type inequalities is present, which are constructed directly from the "standard" Bell inequalities involving two dichotomic observables per site. It is shown that the inequalities are violated by all the generalized Greenberger-Horne-Zeilinger states of multiqubits. Remarkably, our new inequalities can provide stronger non-locality tests in a sense that the local reality inequalities are exponentially stronger than the corresponding multipartite separability inequalities. This reveals that the exponential violation of local realism by separable states is an interesting consequence of quantum fluctuation of multipartite systems.
A proof of Bell's theorem without inequalities valid for both inequivalent classes of three-qubit entangled states under local operations assisted by classical communication, namely Greenberger-Horne-Zeilinger (GHZ) and W,W,W, is described. This proof leads to a Bell inequality that allows more conclusive tests of Bell's theorem for three-qubit systems. Another Bell inequality involving both tri- and bipartite correlations is introduced which illustrates the different violations of local realism exhibited by the GHZ and WWW states.
We present a brief historical introduction to the topic of Bell's theorem. Next we present the surprising features of the three particle Greenberger-Horne-Zeilinger (GHZ) states. Finally we shall present a method of analysis of the GHZ correlations, which is based on a numerical approach, which is effectively equivalent to the full set of Bell inequalities for correlation functions for the given problem. The aim of our numerical approach is to answer the following question. Do additional possible local settings lead for the GHZ states to more pronounced violation of local realism (measured by the resistance of the quantum nature of the correlations with respect ``white'' noise admixtures)?