Sum-of-squares decompositions for a family of CHSH-like inequalities and their application to self-testing

TL;DR

This paper develops sum-of-squares decompositions for tilted CHSH Bell inequalities, providing tight bounds and algebraic relations that enable robust self-testing of partially entangled two-qubit states.

Contribution

It introduces new SOS decompositions for tilted CHSH inequalities and demonstrates their use in robust self-testing, correcting previous flaws in the methodology.

Findings

Tight upper bounds on quantum values of tilted CHSH inequalities.

Algebraic relations characterizing optimal quantum states and observables.

Robust self-testing of partially entangled two-qubit states.

Abstract

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.