More nonlocality with less entanglement

Recent numerical investigations suggest that the I3322 inequality, arguably the simplest extremal Bell inequality after the CHSH inequality, has a very rich structure in terms of the entangled states and measurements that maximally violate it. Here we show that for this inequality the maximally entangled state of any dimension achieves the same violation than just a single EPR pair. In contrast, stronger violations can be achieved using higher dimensional states which are less entangled. This shows that the maximally entangled state is not the most nonlocal resource, even when one restricts attention to the most simple extremal Bell inequalities.