Tsirelson bounds for generalized Clauser-Horne-Shimony-Holt inequalities

By Stephanie Wehner. 2005

138 citations recorded by the source corpus.

Quantum theory imposes a strict limit on the strength of nonlocal correlations. It only allows for a violation of the Clauser, Horne, Shimony, and Holt (CHSH) inequality up to the value 2{radical}(2), known as Tsirelson's bound. In this paper, we consider generalized CHSH inequalities based on many measurement settings with two possible measurement outcomes each. We demonstrate how to prove Tsirelson bounds for any such generalized CHSH inequality using semidefinite programming. As an example, we show that for any shared entangled state and observables X{sub 1},...,X{sub n} and Y{sub 1},...,Y{sub n} with eigenvalues {+-}1 we have | + + + +{center_dot}{center_dot}{center_dot}+ - |{<=}2n cos[{pi}/(2n)]. It is well known that there exist observables such that equality can be achieved. However, we show that these are indeed optimal. Our approach can easily be generalized to other inequalities for such observables.

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

View Paper: https://www.semanticscholar.org/paper/18e63d6c68eccdb6fd3f839332bb9e6c8758fe04

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In Atomic and Molecular Physics, and Optics

Citations in the corpus

A glance beyond the quantum model (2009, background): “The fact that the matrix on the left hand side of equation (C1) is positive semidefinite, implies that there exists a quantum mechanical system able to reproduce the correlators EXY [24].”
TSIRELSON'S PROBLEM AND KIRCHBERG'S CONJECTURE (2010, background): “from the following two remarks which sketch how two other important results on quantum correlations naturally ﬁt into our formalism. Remark 3.5 (The semideﬁnite hierarchy). Building on work of Wehner [Weh06], Navascu´es, Pironio and Ac´ın [NPA08] have developed a hierarchy of semideﬁnite programs ch”
Quantum common causes and quantum causal models (2016, methodology): “By adapting these new techniques to the formalism we present here, one could perhaps systematically derive bounds on the quantum correlations achievable in certain quantum causal models, thereby providing a general method of deriving Tsirelsonlike bounds [81,82] for arbitrary causal structures.”
All noncontextuality inequalities for the n-cycle scenario (2012, methodology): “For the case of even n, the proof is based on positive semidefiniteness conditions analogous to those discussed in [29, 30, 43].”
Large Violation of Bell Inequalities with Low Entanglement (2010, background): “ Bell inequalities (related to the study of deciding whether a given probability distribution belongs to the quantum set Q), has captured the interest of many researchers in QIT (see e.g. [46], [24], [72], [54], [55]). On the other hand, since any two-prover one-round game can be seen as a Bell ineq”