Unbounded violations of bipartite Bell Inequalities via Operator Space theory

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 $sqrt{n}$ (up to a logarithmic factor) 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 $L_p$ 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.