Maximal violation of a broad class of Bell inequalities and its implication on self-testing

In quantum information, lifting is a systematic procedure that can be used to derive - when provided with a seed Bell inequality - other Bell inequalities applicable in more complicated Bell scenarios. It is known that the procedure of lifting introduced by Pironio [J. Math. Phys. 46, 062112 (2005)JMAPAQ0022-248810.1063/1.1928727] preserves the facet-defining property of a Bell inequality. Lifted Bell inequalities therefore represent a broad class of Bell inequalities that can be found in all Bell scenarios. Here, we show that the maximal value of any lifted Bell inequality is preserved for both the set of nonsignaling correlations and quantum correlations. Despite the degeneracy in the maximizers of such inequalities, we show that the ability to self-test a quantum state is preserved under these lifting operations. In addition, except for outcome lifting, local measurements that are self-testable using the original Bell inequality - despite the degeneracy - can also be self-tested using any lifted Bell inequality derived therefrom. While it is not possible to self-test all the positive-operator-valued measure elements using an outcome-lifted Bell inequality, we show that partial, but robust self-testing statements on the underlying measurements can nonetheless be made from the quantum violation of these lifted inequalities. We also highlight the implication of our observations on the usefulness of using lifted Bell-like inequalities as a device-independent witnesses for entanglement depth. The impact of the aforementioned degeneracy on the geometry of the quantum set of correlations is briefly discussed.