Quantum non-locality in two-qutrit systems

Recently a new Bell inequality has been introduced \cite{CGLMP,KKCZO} that is strongly resistant to noise for maximally entangled states of two qudits ($d$-dimensional quantum systems). We prove that a larger violation, or equivalently a stronger resistance to noise, is found for a non-maximally entangled state. In the qutrit case, numerical simulations show that this seems to be the maximal value that can be obtained under any Von Neumann measurement. From these results it follows that two Von Neumann measurements per party are not optimal for detecting non-locality. For $d=3,4$, we also point out some connections between this inequality and distillability. Indeed, we demonstrate that any state violating it, with the optimal Von Neumann settings, is distillable.