Tight upper bound of the maximal quantum violation of Gisin's elegant Bell inequality and its application in randomness certification

TL;DR

This paper establishes a precise upper limit for the quantum violation of Gisin's elegant Bell inequality, enabling improved device-independent randomness certification and revealing advantages over other inequalities in certain quantum states.

Contribution

It provides the first tight upper bound for the maximal quantum violation of Gisin's elegant Bell inequality and characterizes conditions for violation across various two-qubit states.

Findings

Tight upper bound for EBI violation derived.

Conditions for violation of EBI in pure and Werner states identified.

EBI shows advantages over CHSH and chained inequalities near maximally entangled states.

Abstract

The violation of a Bell inequality implies the existence of nonlocality, making device-independent randomness certification possible. This paper derives a tight upper bound for the maximal quantum violation of Gisin's elegant Bell inequality (EBI) for arbitrary two-qubit states, along with the constraints required to achieve this bound. This method provides the necessary and sufficient conditions for violating the EBI for several quantum states, including pure two-qubit states and the Werner states. The lower bound of certifiable global randomness is analyzed based on the tight upper bound of the EBI for pure two-qubit states, with a comparison to the Clauser-Horne-Shimony-Holt (CHSH) inequality. The relationship between the noise level and the lower bound of certifiable global randomness with respect to the Werner states is also explored, and the comparisons with both the CHSH inequality and the chained inequality are given. The results indicate that when the state approaches a maximally entangled state within specific quantified ranges, the EBI demonstrates advantages over both the CHSH inequality and the chained inequality, potentially enhancing practical device-independent randomness generation rates.