All Bell inequalities with a quantum advantage in the minimal scenario are self-testing
In the minimal Bell scenario, all extremal quantum correlations are known, but the Tsirelson inequalities that delimit the quantum set remain poorly understood. We study which correlations can maximize a given Tsirelson inequality, and conversely, which inequalities are maximized by a given correlation. We show that every Tsirelson inequality with quantum advantage is maximized by a single, extremal correlation, and thus self-tests the corresponding realization. This characterizes all self-tests based on linear Bell inequalities in this scenario. We also show that at most a two-dimensional family of inequalities can be maximized by a nonlocal extremal correlation. We identify these as polygons for a family of correlations realized by qubit states of arbitrary entanglement. This allows us to deduce that the quantum set is not self-dual.