Nonlocality, manifested by the violation of Bell inequalities, indicates entanglement within a joint quantum system. A natural question is how much entanglement is required for a given nonlocal behavior. Here, we explore this question by quantifying entanglement using a family of generalized Clauser–Horne–Shimony–Holt-type Bell inequalities. Given a Bell-inequality violation, we derive analytical lower bounds on the entanglement of formation, a measure related to entanglement dilution. The bounds also lead to an analytical estimation of the negativity of entanglement. In addition, we consider one-way distillable entanglement tied to entanglement distillation and derive tight numerical estimates. With the additional assumptions of qubit-qubit systems, we find that the relationship between entanglement and measurement incompatibility is not simply a trade-off under a fixed nonlocal behavior. Furthermore, we apply our results to two realistic scenarios—non-maximally entangled and Werner states. We show that one can utilize the nonlocal statistics by optimizing the Bell inequality for better entanglement estimation.

中文翻译：

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纠缠、测量不兼容和非定域性之间的相互作用


非定域性，表现为违反贝尔不等式，表明联合量子系统内存在纠缠。一个自然的问题是给定的非局域行为需要多少纠缠。在这里，我们通过使用一系列广义的 Clauser-Horne-Shimony-Holt 型贝尔不等式来量化纠缠来探索这个问题。考虑到贝尔不等式的违反，我们得出了形成纠缠的分析下界，这是与纠缠稀释相关的度量。这些界限还导致了对纠缠负面性的分析估计。此外，我们考虑与纠缠蒸馏相关的单向可蒸馏纠缠，并得出严格的数值估计。通过量子位-量子位系统的额外假设，我们发现纠缠和测量不兼容之间的关系不仅仅是固定非局域行为下的权衡。此外，我们将我们的结果应用到两个现实场景中——非最大纠缠态和维尔纳态。我们表明，可以通过优化贝尔不等式来利用非局部统计数据以获得更好的纠缠估计。