Wen-Na Zhao, Youwang Xiao, Ming Li, Li Xu and Shao-Ming Fei
{"title":"SOS decomposition for general Bell inequalities in two qubits systems and its application to quantum randomness","authors":"Wen-Na Zhao, Youwang Xiao, Ming Li, Li Xu and Shao-Ming Fei","doi":"10.1088/1402-4896/ad7536","DOIUrl":null,"url":null,"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.","PeriodicalId":20067,"journal":{"name":"Physica Scripta","volume":null,"pages":null},"PeriodicalIF":2.6000,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Physica Scripta","FirstCategoryId":"101","ListUrlMain":"https://doi.org/10.1088/1402-4896/ad7536","RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"PHYSICS, MULTIDISCIPLINARY","Score":null,"Total":0}
引用次数: 0
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.
贝尔非位置性与器件独立量子随机性密切相关。本文提出了一种针对双量子比特系统中一般贝尔不等式的平方和(SOS)分解。利用得到的 SOS 分解,我们可以找到与所考虑的贝尔不等式的最大违反相关的测量算子。我们还通过考虑(广义)克劳瑟-霍恩-希莫尼-霍尔特(CHSH)贝尔不等式、高雅贝尔不等式、吉辛不等式和链式贝尔不等式作为例子来实践 SOS 分解方法。我们推导出了相应的 SOS 分解和导致这些贝尔不等式最大违反值的测量算子,这与之前的结果是一致的。我们利用贝尔不等式的 SOS 分解进一步讨论了与设备无关的量子随机性。我们以最大纠缠态的广义 CHSH 不等式和达到最大违反值的维尔纳态为例。我们利用 SOS 分解得到了最大猜测概率的精确值或下限。对于维尔纳态,当 p 趋于 1 时,下界可以提供量子随机性的更精确估计。
期刊介绍:
Physica Scripta is an international journal for original research in any branch of experimental and theoretical physics. Articles will be considered in any of the following topics, and interdisciplinary topics involving physics are also welcomed:
-Atomic, molecular and optical physics-
Plasma physics-
Condensed matter physics-
Mathematical physics-
Astrophysics-
High energy physics-
Nuclear physics-
Nonlinear physics.
The journal aims to increase the visibility and accessibility of research to the wider physical sciences community. Articles on topics of broad interest are encouraged and submissions in more specialist fields should endeavour to include reference to the wider context of their research in the introduction.