The Von Neumann Inequality for Polynomials in Several Hilbert-Schmidt Operators

Tonge, Andrew

doi:10.1112/jlms/s2-18.3.519

Cited by 21 publications

(23 citation statements)

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“…(14) is called the (s; r ) norm of T , and we write T (s;r ) . The key result is the following generalization of Grothendieck's inequality, which appears explicitly in [53, Corollary 2.5] (see also [11,17,37,69]). …”

Section: Here Both Identifications Are Isometricmentioning

confidence: 99%

“…Since Grothendieck stated his theorem, a lot of effort has been devoted to find suitable multilinear generalizations (see for instance [11,12,17,53,56,69]). However, up until now, the validity of a trilinear Grothendieck's Theorem in the context of operator spaces (and hence in the context of Bell Inequalities) has been open.…”

Section: N (C B(e F)) = C B(e M N (F))mentioning

confidence: 99%

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Unbounded Violation of Tripartite Bell Inequalities

Pérez-Garcı́a

Wolf

Palazuelos

et al. 2008

Commun. Math. Phys.

117

151

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Abstract: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.

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Section: Here Both Identifications Are Isometricmentioning

confidence: 99%

Section: N (C B(e F)) = C B(e M N (F))mentioning

confidence: 99%

Unbounded Violation of Tripartite Bell Inequalities

Pérez-Garcı́a

Wolf

Palazuelos

et al. 2008

Commun. Math. Phys.

117

151

View full text Add to dashboard Cite

show abstract

“…As mentioned before, there are several generalizations of Grothendieck's matrix inequality ( [3], [19]). Next we state one more such generalization, apparently new.…”

Section: Multiple Summing Operators On L 1 Spacesmentioning

confidence: 99%

“…In Section 4 we generalize Krivine's version of Grothendieck's inequality. As an application we give a simpler proof of one of the multilinear Grothendieck's inequalities stated in [19]. Our proof works for both the real and the complex case and it shows that the constant involved, K n G , is optimal, improving therefore the previous proofs (Tonge's proof works only in the complex case and gives a worse constant and Carne's proof, although it gives the same constant, is not as simple and does not allow to obtain easily the optimality of the constant).…”

Section: Introduction and Notationmentioning

confidence: 99%