Abstract
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.
Highlights
This article gives an elementary exploration of Bell’s demonstration that no classical theory can reproduce the results of quantum mechanics
This article reviews the predictions of quantum mechanics (QM) for one- and two-particle Stern–Gerlach experiments and frames Bell’s results, which rule out hidden-variable alternatives to QM, as attempts by a skeptical Eve to fool Alice and Bob with classical probability mixtures of non-entangled quantum states and a classical hidden-variable theory
VII) closely follows that of Mermin,3 in turn based on a version of Bell due to Bohm
Summary
This article gives an elementary exploration of Bell’s demonstration that no classical theory can reproduce the results of quantum mechanics. VII) closely follows that of Mermin, in turn based on a version of Bell due to Bohm.4 In this Bohm–Mermin formulation, Bell’s famous “inequalities” on correlations among separated observations are equivalently replaced by explicit “programs” for supposed hidden variable theories. The most general state of a single qubit can be written in terms of three Euler angles, 0 a p and 0 b; c 2p as. The most general three-dimensional rotation of an angle h around an axis with unit vector n can be written as the real, orthogonal rotation matrix, Am. J. QM asserts that, a spatial rotation ðh; nÞ in Eq (6) induces a spin transformation in Eq (4) with the same ðh; nÞ This is a deep property of spin-1/2 particles and is possible only because the structures of the continuous groups SU(2) and SO(3) are the same. An accessible proof of this is given in Appendix A
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