Investigations of the boundary of the quantum correlation set have gained increased attention in recent years. This is done through the derivation of quantum Bell inequalities, which are related to Tsirelson's problem and have significant applications in device-independent (DI) information processing. However, determining quantum Bell inequalities is a notoriously difficult task and only isolated examples are known. In this paper, we present families of (almost-)quantum Bell inequalities and highlight four foundational and DI applications. Firstly, it is known that quantum correlations on the non-signaling boundary are of crucial importance in the task of DI randomness extraction from weak sources. In the practical Bell scenario of two players with two $k$-outcome measurements, we derive quantum Bell inequalities that demonstrate a separation between the quantum boundary and certain portions of the boundaries of the no-signaling polytope of dimension up to $4k-8$, extending previous results from nonlocality distillation and the collapse of communication complexity. Secondly as an immediate by-product, we give a general proof of Aumann’s Agreement theorem for quantum systems as well as the almost-quantum correlations, which implies Aumann’s agreement theorem is a reasonable physical principle in the context of epistemics to pick out both quantum theory and almost-quantum correlations from general no-signaling theories. Thirdly, we present a family of quantum Bell inequalities in the two players with $m$ binary measurements scenarios, that we prove serve to self-test the two-qubit singlet and the corresponding $2m$ measurements. Interestingly, this claim generalizes the result for $m=2$ discovered by Tsirelson-Landau-Masanes and shows an improvement over the state-of-the-art Device-Independent Randomness-Amplification (DIRA). Lastly, we use our quantum Bell inequalities to derive the general form of the principle of no advantage in nonlocal computation, which is an information-theoretic principle that serves to characterize the quantum correlation set.

中文翻译：

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（差不多-）量子贝尔不等式和独立于器件的应用


近年来，对量子相关集边界的研究越来越受到关注。这是通过推导量子贝尔不等式来实现的，该不等式与 Tsirelson 问题有关，并在设备无关 （DI） 信息处理中具有重要应用。然而，确定量子贝尔不等式是一项众所周知的困难任务，而且只有孤立的例子是已知的。在本文中，我们介绍了（几乎）量子贝尔不等式的族，并重点介绍了四个基础和 DI 应用。首先，众所周知，非信号边界上的量子相关性在从弱源中提取 DI 随机性的任务中至关重要。在两个玩家具有两个 $k$ 结果测量的实际 Bell 场景中，我们推导出量子 Bell 不等式，证明量子边界与维度高达 $4k-8$ 的无信号多位体边界的某些部分之间的分离，扩展了先前来自非局域性蒸馏和通信复杂性崩溃的结果。其次，作为一个直接的副产品，我们给出了量子系统的奥曼协议定理以及近量子相关性的一般证明，这意味着奥曼协议定理在认识论的背景下是一个合理的物理原理，可以从一般的无信号理论中挑选出量子理论和近量子相关性。第三，我们在两个具有 $m$ 二进制测量情景的玩家中提出了一系列量子贝尔不等式，我们证明这些不等式可用于自检两个量子比特单重态和相应的 $2m$ 测量值。 有趣的是，这一说法概括了 Tsirelson-Landau-Masanes 发现的 $m=2$ 的结果，并显示出对最先进的设备独立随机性放大 （DIRA） 的改进。最后，我们使用量子贝尔不等式来推导出非局部计算中无优势原则的一般形式，这是一个用于描述量子相关集的信息论原理。