Multipartite Bell-type inequalities are derived for general systems. They involve up to eight observables with arbitrary spectra on each site. These inequalities are closely related to the algebras of quaternions and octonions.

arXiv:0803.0913Physical Review ASep 4, 200811 citations

Quantum Information and Computation

Bell inequalities from multilinear contractions

Alejo Salles, Daniel Cavalcanti, Antonio Acín, David Pérez-Garcı́a, Micheal M. Wolf

Published Jul 1, 2010• 3 citations

Publisher

Abstract

DOI: 10.5555/2011373.2011382

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Bell inequalities from multilinear contractions

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n

-partite scenarios to which Bell inequalities are standardly applied, but also Kochen-Specker configurations, and many other examples. There is much current work on experimental tests for contextuality. Our approach directly yields, in a systematic fashion, testable inequalities for a very general notion of contextuality. There has been much work on obtaining proofs of Bell's theorem ``without inequalities'' or ``without probabilities.'' These proofs are seen as being in a sense more definitive and logically robust than the inequality-based proofs. On the hand, they lack the fault-tolerant aspect of inequalities. Our approach reconciles these aspects, and in fact shows how the logical robustness can be converted into systematic, general derivations of inequalities with provable violations. Moreover, the kind of strong non-locality or contextuality exhibited by the GHZ argument or by Kochen-Specker configurations can be shown to lead to maximal violations of the corresponding logical Bell inequalities. Thus the qualitative and the quantitative aspects are combined harmoniously.

(2,2,2)

scenario, namely two parties performing two binary outcome measurements. However, beyond this setting, relatively few Bell inequalities are known, and the situation becomes particularly scarce in scenarios involving a greater number of outcomes. Here, we consider the

(2,2,3)

scenario, or two parties performing two three-outcome measurements, and characterise all Bell inequalities that can arise from the simplest sum-of-squares decomposition and are maximally violated by the maximally entangled state of local dimension three. We then utilise them to self-test this state, along with a class of three-outcome measurements.