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We present a novel tight bound on the quantum violations of the CGLMPinequality in the case of infinitely many outcomes. Like in the case ofTsirelson's inequality the proof of our new inequality does not require anyassumptions on the dimension of the Hilbert space or kinds of operatorsinvolved. However, it is seen that the maximal violation is obtained by theconjectured best measurements and a pure, but not maximally entangled, state.We give an approximate state which, in the limit where the number of outcomestends to infinity, goes to the optimal state for this setting. This state mightbe potentially relevant for experimental verifications of Bell inequalitiesthrough multi-dimenisonal entangled photon pairs.

A tight Tsirelson inequality for infinitely many outcomes