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In quantum information, lifting is a systematic procedure that can be used toderive---when provided with a seed Bell inequality---other Bell inequalitiesapplicable in more complicated Bell scenarios. It is known that the procedureof lifting introduced by Pironio [J. Math. Phys. A 46, 062112 (2005)] preservesthe facet-defining property of a Bell inequality. Lifted Bell inequalitiestherefore represent a broad class of Bell inequalities that can be found in{\em all} Bell scenarios. Here, we show that the maximal value of {\em any}lifted Bell inequality is preserved for both the set of nonsignalingcorrelations and quantum correlations. Despite the degeneracy in the maximizersof such inequalities, we show that the ability to self-test a quantum state ispreserved under these lifting operations. In addition, except foroutcome-lifting, local measurements that are self-testable using the originalBell inequality---despite the degeneracy---can also be self-tested using {\emany} lifted Bell inequality derived therefrom. While it is not possible toself-test {\em all} the positive-operator-valued measure elements using anoutcome-lifted Bell inequality, we show that partial, but robust self-testingstatements on the underlying measurements can nonetheless be made from thequantum violation of these lifted inequalities. We also highlight theimplication of our observations on the usefulness of using lifted Bell-likeinequalities as a device-independent witnesses for entanglement depth. Theimpact of the aforementioned degeneracy on the geometry of the quantum set ofcorrelations is briefly discussed.

Maximal Violation of a Broad Class of Bell Inequalities and Its Implications on Self-Testing