The concept of self-testing (or rigidity) refers to the fact that for certain Bell inequalities the maximal violation can be achieved in an essentially unique manner. In this work we present a family of Bell inequalities which are maximally violated by multiple inequivalent quantum realizations. We completely characterize the quantum realizations achieving the maximal violation and we show that each of them requires a maximally entangled state of two qubits. This implies the existence of a new, weak form of self-testing in which the maximal violation allows us to identify the state, but does not fully determine the measurements. From the geometric point of view the set of probability points that saturate the quantum bound is a line segment. We then focus on a particular member of the family and show that the self-testing statement is robust, i.e., that observing a nonmaximal violation allows us to make a quantitative statement about the unknown state. To achieve this we present a new construction of extraction channels and analyze their performance. For completeness we provide two independent approaches: analytical and numerical. The noise robustness, i.e., the amount of white noise at which the bound becomes trivial, of the analytical bound is rather small ($\approx 0.06\%$), but the numerical method takes us into an experimentally relevant regime ($\approx 5\%$). We conclude by investigating the amount of randomness that can be certified using these Bell violations. Perhaps surprisingly, we find that the qualitative behavior resembles the behavior of rigid inequalities such as the Clauser-Horne-Shimony-Holt inequality. This shows that rigidity is not strictly necessary for device-independent applications.

中文翻译：

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自我测试形式薄弱

自检（或刚性）的概念是指这样的事实，即对于某些Bell不等式，可以以本质上唯一的方式实现最大违规。在这项工作中，我们提出了一个贝尔不等式族，它被多个不等价的量子实现最大程度地破坏了。我们完全刻画了实现最大违背的量子实现，并表明它们每个都需要两个量子位的最大纠缠态。这意味着存在一种新的，较弱的自检形式，其中最大的违例允许我们识别状态，但不能完全确定测量结果。从几何角度来看，使量子界饱和的几组概率点是一条线段。然后，我们着眼于家庭中的某个特定成员，并表明自测陈述是可靠的，即 观察到非最大违规可以使我们对未知状态做出定量陈述。为此，我们提出了一种提取通道的新结构并分析了它们的性能。为了完整起见，我们提供了两种独立的方法：分析方法和数值方法。解析范围的噪声鲁棒性（即边界变得微不足道的白噪声量）很小（$\approx 0.06％$），但数值方法将我们带入了实验相关的区域（$\approx 5％$）。通过调查可以使用这些违反贝尔的行为证明随机性的数量来得出结论。也许令人惊讶的是，我们发现定性行为类似于刚性不等式的行为，例如Clauser-Horne-Shimony-Holt不等式。这表明，对于与设备无关的应用，并非严格要求刚性。