  We have determined numerically the maximum quantum violation of over 100
tight bipartite Bell inequalities with two-outcome measurements by each party
on systems of up to four dimensional Hilbert spaces. We have found several
cases, including the ones where each party has only four measurement choices,
where two dimensional systems, i.e., qubits are not sufficient to achieve
maximum violation. In a significant proportion of those cases when qubits are
sufficient, one or both parties have to make trivial, degenerate 'measurements'
in order to achieve maximum violation. The quantum state corresponding to the
maximum violation in most cases is not the maximally entangled one. We also
obtain the result, that bipartite quantum correlations can always be reproduced
by measurements and states which require only real numbers if there is no
restriction on the size of the local Hilbert spaces. Therefore, in order to
achieve maximum quantum violation on any bipartite Bell inequality (with any
number of settings and outcomes), there is no need to consider complex Hilbert
spaces.
