Unbounded Violation of Tripartite Bell Inequalities
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 GreenbergerHorneZeilinger (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 longstanding open problem in the context of Banach algebras.
 Publication:

Communications in Mathematical Physics
 Pub Date:
 April 2008
 DOI:
 10.1007/s0022000804184
 arXiv:
 arXiv:quantph/0702189
 Bibcode:
 2008CMaPh.279..455P
 Keywords:

 Quantum Physics;
 Mathematical Physics
 EPrint:
 Substantial changes in the presentation to make the paper more accessible for a nonspecialized reader