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arXiv:1302.6698 · Physical Review A

Compact Bell inequalities for multipartite experiments

Yu-Chun Wu, Marek Żukowski, Jing‐Ling Chen, Guang‐Can Guo

Published Aug 29, 2013• 11 citations
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Abstract

A method for construction of multipartite Clauser-Horne-Shimony-Holt- (CHSH-) type Bell inequalities, for the case of local binary observables, is presented. The standard CHSH-type Bell inequalities can be obtained as special cases. An iterative method for deriving $(N+1)$-partite CHSH-type Bell inequalities from $N$-partite ones is derived. As an application, compact Bell inequalities for eight observers involving just four correlation functions are proposed. They require much less experimental effort than standard methods and thus are experimentally friendly in multiphoton experiments.

DOI
10.1103/physreva.88.022126

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A method for construction of multipartite Clauser-Horne-Shimony-Holt- (CHSH-) type Bell inequalities, for the case of local binary observables, is presented. The standard CHSH-type Bell inequalities can be obtained as special cases. An iterative method for deriving (N+1)(N+1)(N+1)-partite CHSH-type Bell inequalities from NNN-partite ones is derived. As an application, compact Bell inequalities for eight observers involving just four correlation functions are proposed. They require much less experimental effort than standard methods and thus are experimentally friendly in multiphoton experiments.
We construct a simple algorithm to generate any Clauser-Horne-Shimony-Holt- (CHSH-) type Bell inequality involving a party with two local binary measurements from two CHSH-type inequalities without this party. The algorithm readily generalizes to situations where the additional observer uses three measurement settings. There, each inequality involving the additional party is constructed from three inequalities with this party excluded. With this generalization at hand, we construct and analyze a class of symmetric inequalities for four observers and three experimental settings per observer.
Based on Clauser-Horne-Shimony-Holt inequality, we show a fruitful method to exploit Bell inequalities for multipartite qubit systems. These Bell inequalities are designed with a simpler architecture tailored to experimental demonstration. With the point of view of the stabilizer formalism, we suggest a method to investigate quantum nonlocality for multipartite systems. Under the optimal setting we derive a set of compact Mermin-type inequalities and then discuss quantum violations for generalized Greenberger-Horne-Zeilinger (GGHZ) states and two kinds of mixed states. Also, as an example, we reveal relationship between quantum nonlocality and four-partite entanglement for four-qubit GGHZ states.
A technique, which we call homogenization, is applied to transform Clauser and Horne- (CH)-type Bell inequalities that contain lower-order correlations into CH, Shimony, and Holt- (CHSH)-type Bell inequalities, which are defined for highest-order correlation functions. A homogenization leads to inequalities involving more settings, that is, a choice of one more observable is possible for each party. We show that this technique preserves the tightness of Bell inequalities: A homogenization of a tight CH-type Bell inequality is still a tight CHSH-type Bell inequality. As an example, we obtain $3\ifmmode\times\else\texttimes\fi{}3\ifmmode\times\else\texttimes\fi{}3$ CHSH-type Bell inequalities by the homogenization of $2\ifmmode\times\else\texttimes\fi{}2\ifmmode\times\else\texttimes\fi{}2$ CH-type Bell inequalities derived by Sliwa [Phys. Lett. A 317, 165 (2003)].
A number of papers have suggested that it is inappropriate to combine data from different experiments when undertaking experimental tests of Bell's inequalities. It has been suggested that a correct analysis, using a single probability space, leads to inequalities which are not violated by experiment. If correct, this would be contrary to the normal interpretation of such experimental data. However, in this note, a generalised Clauser-Horne-Shimony-Holt (CHSH) inequality is derived for a system of four experiments constructed on a single probability space which combines the data from the four experiments. It is shown that this leads to the standard CHSH inequality which is normally used to interpret experimental data. Thus the commonly accepted conclusion that experimental violations of Bell's inequality imply that local realistic models are inconsistent with the predictions of quantum mechanics has not been challenged.
We generalize the correlation functions of the Clauser-Horne-Shimony-Holt (CHSH) inequality to multipartite ddd-dimensional systems. All the Bell inequalities based on this generalization take the same simple form as the CHSH inequality. For small systems, numerical results show that the new inequalities are tight and we believe this is also valid for higher-dimensional systems. Moreover, the new inequalities are relevant to the previous ones and, for bipartite systems, our inequality is equivalent to the Collins-Gisin-Linden-Masser-Popescu inequality.
The Bell-Clauser-Horne-Shimony-Holt (BCHSH) inequality, which is proven in the context of the local hidden-variable theory, has been used as a test to reveal the failure of the hidden-variable theory and to prove the validity of the quantum theory. We note that the violation of the BCHSH inequality is caused by the noncommutativity of quantum observables and find a systematic method of constructing generalizations of the BCHSH inequality. This method is used to invent a new quantity, which is defined in a two-qubit system and satisfies a new type of inequality. This provides a fair test of the quantum theory. Remaining problems are also discussed.