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arXiv:2504.13427 · Laser Physics Letters

Tight upper bound and monogamy relation for the maximum quantum value of the parity-CHSH inequality and applied to device-independent randomness

Guannan Zhang, Jiamin Xu, Ming Li, Shu‐Qian Shen, Lei Li, Shao-Ming Fei

Published Apr 16, 2025• 1 citation
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Abstract

Abstract Based on the violation of Bell inequalities, we can verify quantum random numbers by examining the correlation between device inputs and outputs. In this paper, we derive the maximum quantum value of the parity-CHSH inequality for a three-qubit system, establishing a tight upper bound applicable to any quantum state. Simultaneously, the necessary constraints for achieving saturation are analyzed. Utilizing this method, we present necessary and sufficient conditions for certain states to violate the parity-CHSH inequality. Building upon our proposal, the relationship between the noise parameter and the certifiable randomness in a bipartite entangled state is probed. Furthermore, we derive a monogamy relationship between the average values of the parity-CHSH inequality associated with the reduced three-qubit density matrices of GHZ-class states comprising four qubits.

DOI
10.1088/1612-202x/adca9a

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Abstract Based on the violation of Bell inequalities, we can verify quantum random numbers by examining the correlation between device inputs and outputs. In this paper, we derive the maximum quantum value of the parity-CHSH inequality for a three-qubit system, establishing a tight upper bound applicable to any quantum state. Simultaneously, the necessary constraints for achieving saturation are analyzed. Utilizing this method, we present necessary and sufficient conditions for certain states to violate the parity-CHSH inequality. Building upon our proposal, the relationship between the noise parameter and the certifiable randomness in a bipartite entangled state is probed. Furthermore, we derive a monogamy relationship between the average values of the parity-CHSH inequality associated with the reduced three-qubit density matrices of GHZ-class states comprising four qubits.
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.
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.
Two parties sharing entangled quantum systems can generate correlations that cannot be produced using only shared classical resources. These nonlocal correlations are a fundamental feature of quantum theory but also have practical applications. For instance, they can be used for device-independent random number generation, whose security is certified independently of the operations performed inside the devices. The amount of certifiable randomness that can be generated from some given nonlocal correlations is a key quantity of interest. Here, we derive tight analytic bounds on the maximum certifiable randomness as a function of the nonlocality as expressed using the Clauser-Horne-Shimony-Holt (CHSH) value. We show that for every CHSH value greater than the local value (2) and up to 3sqrt[3]/2≈2.598 there exist quantum correlations with that CHSH value that certify a maximal two bits of global randomness. Beyond this CHSH value the maximum certifiable randomness drops. We give a second family of Bell inequalities for CHSH values above 3sqrt[3]/2, and show that they certify the maximum possible randomness for the given CHSH value. Our work hence provides an achievable upper bound on the amount of randomness that can be certified for any CHSH value. We illustrate the robustness of our results, and how they could be used to improve randomness generation rates in practice, using a Werner state noise model.
Bell nonlocality provides a device-independent (DI) way to certify quantum randomness, based on which true random numbers can be extracted from the observed correlations without detail characterizations on devices for quantum state preparation and measurement. However, the efficiency of current strategies for DI randomness certification is still heavily constrained when it comes to non-maximal Bell values, especially for multiple parties. Here, we present a family of multipartite Bell inequalities that allows to certify optimal quantum randomness and self-test GHZ (Greenberger-Horne-Zeilinger) states, which are inspired from the stabilizer group of the GHZ state. Due to the simple representation of stabilizer group for GHZ states, this family of Bell inequalities is of simple structure and can be easily expanded to more parties. Compared with the Mermin-type inequalities, this family of Bell inequality is more efficient in certifying quantum randomness when non-maximal Bell values achieved. Meanwhile, the general analytical upper bound for the Holevo quantity is presented, and achieves better performance compared with the MABK (Mermin-Ardehali-Belinskii-Klyshko) inequality, Parity-CHSH (Clauser-Horne-Shimony-Holt) inequality and Holz inequality at N=3N=3N=3, which is of particular interests for experimental researches on DI quantum cryptography in quantum networks.
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.
We investigate the relationship between CHSH inequality violation and quantum Fisher information (QFI) maximized over local operations and classical communication (LOCC) for two-qubit systems. Through systematic analysis of 1000 random states, we discover that no state with Bell parameter M(ρ) exceeding maximized QFI violates the CHSH inequality, suggesting QFImax serves as a necessary condition. States partition into three classes: (1) M(ρ) > QFImax with CHSH violation (23%), (2) M(ρ) > QFImax without violation (41%), and (3) M(ρ) < QFImax without violation (36%). We identify relative entropy of entanglement thresholds (0.14 and 0.26) separating regions of guaranteed violation, ambiguous behavior, and no violation. Furthermore, 98% of states exhibit QFI values that cross the separability bound under local rotations. These results provide efficient pre-screening criteria for device-independent quantum protocols and reveal deep connections between quantum metrology and Bell nonlocality.