Sum-of-squares decompositions for a family of Clauser-Horne-Shimony-Holt-like inequalities and their application to self-testing

We introduce two families of sum-of-squares (SOS) decompositions for the Bell operators associated with the tilted Clauser-Horne-Shimony-Holt (CHSH) expressions introduced in Ac\'{}n et al. [Phys. Rev. Lett. 108, 100402 (2012)]. These SOS decompositions provide tight upper bounds on the maximal quantum value of these Bell expressions. Moreover, they establish algebraic relations that are necessarily satisfied by any quantum state and observables yielding the optimal quantum value. These algebraic relations are then used to show that the tilted CHSH expressions provide robust self-tests for any partially entangled two-qubit state. This application to self-testing follows closely the approach of Yang and Navascu\'es [Phys. Rev. A 87, 050102(R) (2013)], where we identify and correct two nontrivial flaws.