(Nearly) optimal P values for all Bell inequalities
(Nearly) optimal P values for all Bell inequalities
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DOI：
10.1038/npjqi.2016.26
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发表时间：
2016-10-18
影响因子：
7.6
通讯作者：
Wehner, Stephanie
中科院分区：
文献类型：
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作者：
Elkouss, David;Wehner, Stephanie
进行贝尔测试的一个关键目标是量化针对局部隐变量模型 (LHVM) 的统计证据，因为我们在任何实验中只能收集有限数量的试验。因此，统计证据的概念是在假设检验的框架中制定的，其中零假设是实验可以通过 LHVM 进行描述。拒绝 LHVM 零假设的统计置信度通过所谓的 P 值进行量化，其中较小的 P 值意味着较高的置信度。如果试验数量很少，或者贝尔违规率非常低，那么建立良好的统计证据尤其具有挑战性。在这里，我们推导出一大类贝尔不等式的最优 P 值。更重要的是，我们获得了所有贝尔不等式的 P 值的非常尖锐的上限。这些值很容易从实验数据中计算出来，即使我们允许设备中存在任意内存，这些值也是有效的。我们的分析能够处理不完美的随机数生成器和事件就绪方案，即使这样的方案可以创建不同类型的纠缠态。最后，我们回顾了健全数据收集的要求，以及结合独立实验 P 值的方法。这里讨论的方法并不特定于贝尔不等式。例如，它们还可以应用于认证随机性的研究或非上下文性的测试。
A key objective in conducting a Bell test is to quantify the statistical evidence against a local-hidden variable model (LHVM) given that we can collect only a finite number of trials in any experiment. The notion of statistical evidence is thereby formulated in the framework of hypothesis testing, where the null hypothesis is that the experiment can be described by an LHVM. The statistical confidence with which the null hypothesis of an LHVM is rejected is quantified by the so-called P value, where a smaller P value implies higher confidence. Establishing good statistical evidence is especially challenging if the number of trials is small, or the Bell violation very low. Here, we derive the optimal P value for a large class of Bell inequalities. What is more, we obtain very sharp upper bounds on the P value for all Bell inequalities. These values are easily computed from the experimental data, and are valid even if we allow arbitrary memory in the devices. Our analysis is able to deal with imperfect random number generators, and event-ready schemes, even if such a scheme can create different kinds of entangled states. Finally, we review requirements for sound data collection, and a method for combining P values of independent experiments. The methods discussed here are not specific to Bell inequalities. For instance, they can also be applied to the study of certified randomness or to tests of noncontextuality.