SOS decomposition for general Bell inequalities in two qubits systems and its application to quantum randomness

TL;DR

This paper introduces a sum-of-squares (SOS) decomposition method for general Bell inequalities in two-qubit systems, enabling the derivation of measurement operators for maximal violations and advancing device-independent quantum randomness estimation.

Contribution

It develops a general SOS decomposition framework for Bell inequalities in two-qubit systems and applies it to derive measurement operators and bounds on quantum randomness.

Findings

Derived SOS decompositions for several Bell inequalities.

Identified measurement operators for maximum Bell violation.

Provided precise bounds on quantum randomness for Werner states.

Abstract

Bell non-locality is closely related with device independent quantum randomness. In this paper, we present a kind of sum-of-squares (SOS) decomposition for general Bell inequalities in two qubits systems. By using the obtained SOS decomposition, we can then find the measurement operators associated with the maximal violation of considered Bell inequality. We also practice the SOS decomposition method by considering the (generalized) Clauser-Horne-Shimony-Holt (CHSH) Bell inequality, the Elegant Bell inequality, the Gisin inequality and the Chained Bell inequality as examples. The corresponding SOS decompositions and the measurement operators that cause the maximum violation values of these Bell inequalities are derived, which are consistent with previous results. We further discuss the device independent quantum randomness by using the SOS decompositions of Bell inequalities. We take the generalized CHSH inequality with the maximally entangled state and the Werner state that attaining the maximal violations as examples. Exact value or lower bound on the maximal guessing probability using the SOS decomposition are obtained. For Werner state, the lower bound can supply a much precise estimation of quantum randomness when $p$ tends to $1$.