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 $3\sqrt{3}/2\approx 2.598$ 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 $3\sqrt{3}/2$, 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.

中文翻译：

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与设备无关的随机性和非局域性之间权衡的紧密分析界限

共享纠缠量子系统的两方可以产生仅使用共享经典资源无法产生的相关性。这些非局部相关性是量子理论的一个基本特征，但也有实际应用。例如，它们可用于与设备无关的随机数生成，其安全性独立于设备内部执行的操作进行认证。可以从某些给定的非局部相关性中产生的可证明随机性的数量是一个关键的关注量。在这里，我们推导出最大可证明随机性的严格分析界限，作为使用 Clauser-Horne-Shimony-Holt (CHSH) 值表示的非局域性的函数。我们证明，对于每个大于局部值 (2) 且最多为$3\sqrt{3}/2\approx 2.598$与该 CHSH 值存在量子相关性，这证明了最大两位全局随机性。超过这个 CHSH 值，最大可证明随机性下降。对于上面的 CHSH 值，我们给出了第二个 Bell 不等式族$3\sqrt{3}/2$，并表明它们证明了给定 CHSH 值的最大可能随机性。因此，我们的工作为任何 CHSH 值提供了一个可实现的随机量上限。我们使用 Werner 状态噪声模型说明了我们的结果的稳健性，以及如何在实践中使用它们来提高随机性生成率。