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The I3322 inequality is the simplest bipartite two-outcome Bell inequalitybeyond the Clauser-Horne-Shimony-Holt (CHSH) inequality, consisting of threetwo-outcome measurements per party. In case of the CHSH inequality the maximalquantum violation can already be attained with local two-dimensional quantumsystems, however, there is no such evidence for the I3322 inequality. In thispaper a family of measurement operators and states is given which enables us toattain the largest possible quantum value in an infinite dimensional Hilbertspace. Further, it is conjectured that our construction is optimal in the sensethat measuring finite dimensional quantum systems is not enough to achieve thetrue quantum maximum. We also describe an efficient iterative algorithm forcomputing quantum maximum of an arbitrary two-outcome Bell inequality in anygiven Hilbert space dimension. This algorithm played a key role to obtain ourresults for the I3322 inequality, and we also applied it to improve on ourprevious results concerning the maximum quantum violation of several bipartitetwo-outcome Bell inequalities with up to five settings per party.

Maximal violation of the I3322 inequality using infinite dimensional quantum systems