This article reviews the predictions of quantum mechanics (QM) for one- and two-particle Stern–Gerlach experiments and then frames Bell's results, which rule out hidden-variable alternatives to QM, as attempts by a skeptical Eve to fool Alice and Bob with (first example) classical probability mixtures of non-entangled quantum states and (second example) a classical hidden-variable theory. With hidden variables, Eve can succeed when Alice and Bob limit themselves to two Stern–Gerlach directions, but always fails for some choices of three directions. For three random directions, hidden-variable theories are impossible (we show) exactly 2/3 of the time. All the calculations in this article are available in Mathematica and Python files in the supplementary material, allowing readers to experiment with their own variants.

中文翻译：

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Bell、Bohm 和 qubit：EPR 混音

本文回顾了量子力学 (QM) 对单粒子和双粒子 Stern-Gerlach 实验的预测，然后对 Bell 的结果进行了框架化，该结果排除了 QM 的隐藏变量替代方案，因为怀疑的夏娃试图用（第一个例子）非纠缠量子态的经典概率混合和（第二个例子）一个经典的隐藏变量理论。有了隐藏变量，当 Alice 和 Bob 将自己限制在两个 Stern-Gerlach 方向时，Eve 可以成功，但对于三个方向的某些选择总是失败。对于三个随机方向，隐变量理论在 2/3 的时间内是不可能的（我们展示）。本文中的所有计算都可以在补充材料中的 Mathematica 和 Python 文件中找到，让读者可以试验自己的变体。