Large violation of Bell inequalities with low entanglement

In this work we show the existence of violations of general bipartite Bell inequalities of order √ n log n with n inputs, n outputs and n-dimensional Hilbert spaces. Analyzing the construction of the elements involved in this violation one finds that, even though entanglement is necessary to obtain violation of Bell inequalities, the entropy of entanglement of the underlying state is essentially irrelevant in obtaining large violation. We also indicate why the maximally entangled state is a rather poor candidate in producing large violations with arbitrary coefficients. However, we also show that for Bell inequalities with positive coefficients (in particular, games) the maximally entangled state achieves the largest violation up to a logarithmic factor.