Logical Bell inequalities
Logical Bell inequalities
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DOI：
10.1103/physreva.85.062114
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发表时间：
2012-06-20
影响因子：
2.9
通讯作者：
Hardy, Lucien
中科院分区：
文献类型：
--
作者：
Abramsky, Samson;Hardy, Lucien
贝尔不等式在量子非定域和纠缠的研究中起着核心作用，在量子信息中有许多应用。尽管有大量关于贝尔不等式的文献，但要找到一个明确的概念答案并不容易，也不容易找到一个明确的指导原则来解释贝尔不等式是什么，或者如何推导出它们。本文引入了逻辑贝尔不等式的概念，该概念可用于系统地推导各种情况下的可检验不等式。有一个明确的概念原则，建立在纯逻辑一致性条件的基础上，它是逻辑贝尔不等式概念的基础。我们证明了在精确的意义上，所有的贝尔不等式都可以被看作是这种形式。我们的方法很一般。它直接适用于任何可交换可观测集合族。因此，它不仅涵盖了标准应用贝尔不等式的n-部场景，而且还涵盖了Kochen-Specker配置和许多其他示例。目前有很多关于情境性实验测试的工作。我们的方法以一种系统的方式直接产生了可测试的不等式，用于非常一般的情境概念。人们做了很多工作来证明“不需要不等式”或“不需要概率”的贝尔定理。这些证明在某种意义上被视为比基于不等式的证明更明确和逻辑上更可靠。一方面，它们缺乏不平等的容错性。我们的方法调和了这些方面，并且实际上显示了逻辑鲁棒性如何可以转换为具有可证明违反的不等式的系统的一般推导。此外，GHZ论证或Kochen-Specker配置所表现出的那种强非局部性或情境性可以被证明会导致对相应逻辑贝尔不等式的最大违反。从而使定性和定量两个方面和谐地结合起来。
Bell inequalities play a central role in the study of quantum nonlocality and entanglement, with many applications in quantum information. Despite the huge literature on Bell inequalities, it is not easy to find a clear conceptual answer to what a Bell inequality is, or a clear guiding principle as to how they may be derived. In this paper, we introduce a notion of logical Bell inequality which can be used to systematically derive testable inequalities for a very wide variety of situations. There is a single clear conceptual principle, based on purely logical consistency conditions, which underlies our notion of logical Bell inequalities. We show that in a precise sense, all Bell inequalities can be taken to be of this form. Our approach is very general. It applies directly to any family of sets of commuting observables. Thus it covers not only the n-partite scenarios to which Bell inequalities are standardly applied, but also Kochen-Specker configurations, and many other examples. There is much current work on experimental tests for contextuality. Our approach directly yields, in a systematic fashion, testable inequalities for a very general notion of contextuality. There has been much work on obtaining proofs of Bell's theorem "without inequalities" or "without probabilities." These proofs are seen as being in a sense more definitive and logically robust than the inequality-based proofs. On the hand, they lack the fault-tolerant aspect of inequalities. Our approach reconciles these aspects, and in fact shows how the logical robustness can be converted into systematic, general derivations of inequalities with provable violations. Moreover, the kind of strong non-locality or contextuality exhibited by the GHZ argument or by Kochen-Specker configurations can be shown to lead to maximal violations of the corresponding logical Bell inequalities. Thus the qualitative and the quantitative aspects are combined harmoniously.