  We introduce two families of sum-of-squares (SOS) decompositions for the Bell
operators associated with the tilted CHSH expressions introduced in 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 Phys. Rev. A 87, 050102 (2013),
where we identify and correct two non-trivial flaws.
