We design a Bell inequality that is violated by correlations obtained from the ground states of an
XXZ spin-1 chain with on-site anisotropies at the region of phase transition. In order to detect such correlations in spin-1 systems we exploit the formalism of the generalized Bell inequality via the use of multipartite and high-order correlations. We observe a sharp violation in the vicinity of the quantum phase transition between the so-called large-
D and AFM phase. Interestingly, the violation of our Bell inequality is manifested by the change in the
XXZ spin-1 chain ground state to a Greenberger-Horne-Zeilinger-like state in the critical region. Our results provide a characterization of quantum phase transition via the violation of Bell-type constraint by correlations in the
XXZ spin-1 chain with multibody correlations and high-order measurements.
The violation of the Bell inequality not only provides the most radical departure of quantum theory from classical concepts, but also paves the way for applications such as device-independent randomness certification. Here, we derive the tight upper bound of the maximum quantum value for the chained Bell inequality with an arbitrary number of measurements on each party. The constraints where the upper bound saturates are also presented. This method provides us the necessary and sufficient conditions for some quantum states to violate the chained Bell inequality with an arbitrary number of measurements. Based on the tight upper bound we present the lower bounds on the device-independent randomness with respect to the Werner states. In particular, we present lower bounds on the randomness generation rates of the chained Bell inequality for a different number of measurements, which are compared with the family of Bell inequalities proposed by Wooltorton et al. [Phys. Rev. Lett. 129, 150403 (2022)]. Our results show that the chained Bell inequality with three measurements has certain advantages at a low level of noise and could be used to improve randomness generation rates in practice.
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, providing theoretical guidance for experimental device-independent quantum random number generation.