Device-independent protocols based on Bell nonlocality, such as quantum key distribution and randomness generation, must ensure no adversary can have prior knowledge of the measurement outcomes. This requires a measurement independence assumption: that the choice of measurement is uncorrelated with any other underlying variables that influence the measurement outcomes. Conversely, relaxing measurement independence allows for a fully “causal” simulation of Bell nonlocality. We construct the most efficient such simulation, as measured by the mutual information between the underlying variables and the measurement settings, for the Clauser-Horne-Shimony-Holt (CHSH) scenario, and find that the maximal quantum violation requires a mutual information of just $\sim 0.080$ bits. Any physical device built to implement this simulation allows an adversary to have full knowledge of a cryptographic key or “random” numbers generated by a device-independent protocol based on violation of the CHSH inequality. We also show that a previous model for the CHSH scenario, requiring only $\sim 0.046$ bits to simulate the maximal quantum violation, corresponds to the most efficient “retrocausal” simulation, in which future measurement settings necessarily influence earlier source variables. This may be viewed either as an unphysical limitation of the prior model or as an argument for retrocausality on the grounds of its greater efficiency. Causal and retrocausal models are also discussed for maximally entangled two-qubit states, as well as superdeterministic, one-sided, and zigzag causal models.

中文翻译：

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贝尔非本地性的测量依赖成本：因果模型与因果模型

基于贝尔非局部性的与设备无关的协议（例如，量子密钥分配和随机性生成）必须确保没有任何对手能够事先了解测量结果。这就需要一个测量独立性的假设：测量的选择与影响测量结果的任何其他潜在变量都不相关。相反，放松的测量独立性允许对Bell非局部性进行完全“因果”模拟。我们针对Clauser-Horne-Shimony-Holt（CHSH）场景构造了最有效的此类模拟（通过基础变量和测量设置之间的互信息来度量），发现最大的量子违背性要求$〜0.080$位。为实现此模拟而构建的任何物理设备都允许对手全面了解由与设备无关的协议基于违反CHSH不等式产生的加密密钥或“随机”数字。我们还显示了用于CHSH场景的先前模型，仅需要$〜0.046$用于模拟最大量子违规的位对应于最有效的“回因”模拟，其中将来的测量设置必定会影响早期的源变量。这既可以看作是现有模型的非物质限制，也可以看作是因果关系，因为它具有更高的效率。还讨论了最大纠缠的两个量子位状态的因果关系模型和超因果关系模型，以及超确定性，单边和之字形因果关系模型。