Bell inequalities play a key role in certifying quantum properties for device-independent quantum information protocols. It is still a major challenge, however, to devise Bell inequalities tailored for an arbitrary given quantum state. Existing approaches based on sums of squares provide results in this direction, but they are restricted by the necessity of first choosing measurement settings suited to the state. Here, we show how the sum of square property can be enforced for an arbitrary target state by making an appropriate choice of nullifiers, which is made possible by leaving freedom in the choice of measurement. Using our method, we construct simple Bell inequalities for several families of quantum states, including partially entangled multipartite GHZ states and qutrit states. In most cases we are able to prove that the constructed Bell inequalities achieve self-testing of the target state. We also use the freedom in the choice of measurement to self-test partially entangled two-qubit states with a family of settings with two parameters. Finally, we show that some statistics can be self-tested with distinct Bell inequalities, hence obtaining new insight on the shape of the set of quantum correlations.

中文翻译：

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形式平方和的自定义贝尔不等式

贝尔不等式在证明与设备无关的量子信息协议的量子特性方面发挥着关键作用。然而，设计适合任意给定量子态的贝尔不等式仍然是一个重大挑战。基于平方和的现有方法提供了这个方向的结果，但它们受到首先选择适合状态的测量设置的必要性的限制。在这里，我们展示了如何通过适当选择无效器来对任意目标状态强制执行平方和属性，这可以通过保留测量选择的自由度来实现。使用我们的方法，我们为几个量子态族构造了简单的贝尔不等式，包括部分纠缠的多部分 GHZ 态和 qutrit 态。在大多数情况下，我们能够证明构造的贝尔不等式实现了目标状态的自检。我们还利用测量选择的自由度，通过一系列具有两个参数的设置来自测试部分纠缠的两个量子位态。最后，我们表明一些统计数据可以通过不同的贝尔不等式进行自我测试，从而获得关于量子相关性集的形状的新见解。