Communications in Mathematical Physics

, Volume 279, Issue 2, pp 455-486

First online:

Open Access This content is freely available online to anyone, anywhere at any time.

Unbounded Violation of Tripartite Bell Inequalities

  • D. Pérez-GarcíaAffiliated withDepartamento de Análisis Matemático, Universidad Complutense de Madrid Email author 
  • , M. M. WolfAffiliated withMax Planck Institut für Quantenoptik
  • , C. PalazuelosAffiliated withDepartamento de Análisis Matemático, Universidad Complutense de Madrid
  • , I. VillanuevaAffiliated withDepartamento de Análisis Matemático, Universidad Complutense de Madrid
  • , M. JungeAffiliated withDepartment of Mathematics, University of Illinois at Urbana-Champaign


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.