Maximal violation of a bipartite three-setting, two-outcome Bell inequality using infinite-dimensional quantum systems

The ${I}_{3322}$ inequality is the simplest bipartite two-outcome Bell inequality beyond the Clauser-Horne-Shimony-Holt (CHSH) inequality, consisting of three two-outcome measurements per party. In the case of the CHSH inequality the maximal quantum violation can already be attained with local two-dimensional quantum systems; however, there is no such evidence for the ${I}_{3322}$ inequality. In this paper a family of measurement operators and states is given which enables us to attain the maximum quantum value in an infinite-dimensional Hilbert space. Further, it is conjectured that our construction is optimal in the sense that measuring finite-dimensional quantum systems is not enough to achieve the true quantum maximum. We also describe an efficient iterative algorithm for computing quantum maximum of an arbitrary two-outcome Bell inequality in any given Hilbert space dimension. This algorithm played a key role in obtaining our results for the ${I}_{3322}$ inequality, and we also applied it to improve on our previous results concerning the maximum quantum violation of several bipartite two-outcome Bell inequalities with up to five settings per party.