Resumen: In this work we show that bipartite quantum states with local Hilbert space dimension n can violate a Bell inequality by a factor of order Ω(√n∕Log2n) when observables with n possible outcomes are used. A central tool in the analysis is a close relation between this problem and operator space theory and,
in particular, the very recent noncommutative Lp embedding theory.
As a consequence of this result, we obtain better Hilbert space dimension witnesses and quantum violations of Bell inequalities with better resistance to noise.
Resumen: 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 commu-
nication complexity theory. Moreover, we show that the violation obtained from generalized GHZ
states is always bounded so that, in contrast to many other contexts, GHZ states do in this case not
lead to extremal quantum correlations. 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.
