  We present a novel tight bound on the quantum violations of the CGLMP
inequality in the case of infinitely many outcomes. Like in the case of
Tsirelson's inequality the proof of our new inequality does not require any
assumptions on the dimension of the Hilbert space or kinds of operators
involved. However, it is seen that the maximal violation is obtained by the
conjectured best measurements and a pure, but not maximally entangled, state.
We give an approximate state which, in the limit where the number of outcomes
tends to infinity, goes to the optimal state for this setting. This state might
be potentially relevant for experimental verifications of Bell inequalities
through multi-dimenisonal entangled photon pairs.
