Unbounded Violation of Tripartite Bell Inequalities

By David Pérez-García, Michael M. Wolf, Carlos Palazuelos, Ignacio Villanueva, Marius Junge. 2007

132 citations recorded by the source corpus.

We prove that there are tripartite quantum states (constructed from random unitaries) that can lead to arbitrarily large violations of Bell inequalities for dichotomic observables. As a consequence these states can withstand an arbitrary amount of white noise before they admit a description within a local hidden variable model. This is in sharp contrast with the bipartite case, where all violations are bounded by Grothendieck’s constant. We will discuss the possibility of determining the Hilbert space dimension from the obtained violation and comment on implications for communication complexity theory. Moreover, we show that the violation obtained from generalized Greenberger-Horne-Zeilinger (GHZ) states is always bounded so that, in contrast to many other contexts, GHZ states do not lead to extremal quantum correlations in this case. In order to derive all these physical consequences, we will have to obtain new mathematical results in the theories of operator spaces and tensor norms. In particular, we will prove the existence of bounded but not completely bounded trilinear forms from commutative C*-algebras. Finally, we will relate the existence of diagonal states leading to unbounded violations with a long-standing open problem in the context of Banach algebras.

Subfields: Artificial Intelligence, Atomic and Molecular Physics, and Optics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/7e03d7babb7a08ade8a7bbbae186b76b087e881e

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In Artificial Intelligence

In Atomic and Molecular Physics, and Optics

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Citations in the corpus

Entanglement detection (2008, background): “For three parties and for two-outcome observables, inequalities with arbitrarily large violation can be designed [344].”
Large Violation of Bell Inequalities with Low Entanglement (2010, background): “it enriches the theory of multipartite interactive proof systems ([7, 19, 18, 32, 24, 41, 39]), communication complexity (see the recent review [15]); Estimates for the dimension of the underlying Hilbert space ([11, 13, 57, 71, 73]), entangled games ([41, 40]), etc.”
Grothendieck's Theorem, past and present (2011, methodology): “This was recently answered by Marius Junge with Pérez-Garćıa, Wolf, Palazuelos, Villanueva [39].”