Comparison of two bounds of the quantum correlation set

From a geometric viewpoint, quantum nonlocality between two parties is represented as the difference of two convex bodies, namely the sets of possible results of classical and quantum correlation experiments, the latter of which is called the quantum correlation set. Whereas little is known about the quantum correlation set, Tsirelson´s theorem (1980) can be seen as the exact characterization of possible pairwise quantum correlations, where mean values of individual observables are discarded. In this paper, we compare two previously shown bounds of the quantum correlation set in the case where two parties have m and n choices of dichotomic observables, respectively. One bound comes from the direct application of Tsirelson´s theorem and the no-signalling condition. The other bound, recently introduced by Avis, Imai and Ito, refines the application of Tsirelson´s theorem in the previous bound. We show that for any m, n ges 2, this new bound is strictly tighter than the earlier bound. In other words, there are correlations that satisfy Tsirelson´s theorem, but are not realizable in a quantum setting.