Spooky Action at a Distance 
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electron #1:  F+L+R–  F  F  L  L  R  R  
electron #2:  F–L–R+  L  R  F  R  F  L  
spin difference:  2/6  d  d  
Pr(measured spins are different) = 1⁄3 
For example, if in the case above we measured electron #1 on the F axis and electron #2 on the L axis, we would have found that the two spins are different (hence the “d” above), since electron #1 is F+ and electron #2 is L–. If however we measured electron #1 on the F axis and electron #2 on the R axis, we would have found both to be “spin up” (electron #1 is F+ and electron #2 is R+). We can see from the above table that if we randomly picked two different axes, there’s a ⅓ chance of finding different spins. Let’s check all possible spin orientations of the electron pairs:


So no matter which electron pair we get for our experiment, we get this inequality:
Pr(measured spins are different) ≥ 1⁄3.For reasons I’ll explain a little later, the above sort of inequality is called Bell’s inequality. Our inequality is different from the predictions of quantum mechanics, which calculates the probability this way:
Pr(measured spins are different) = cos^{2}(0.5 × angle between #1 and #2)The angle between electrons #1 and #2 will always be either 120° or 240°. Using the above equation both turn out to have a probability of ¼. So who’s right? Quantum mechanics or local hidden variable theory? The answer of course is quantum mechanics.
While our main target was local hidden variables theory, Bell’s inequality can be used to argue for nonlocality in general. One might think the localist could just reject hidden variables theory and locality is saved, right? Well, it’s not quite that simple. Consider the cases where both electrons have their spins measured along the same axis. If observing one electron’s spin doesn’t causally influence the other, how is it that the spins are always opposite of each other regardless of which of the three axis pairs we pick? How does the other electron always “know” which spin is opposite of its partner? One could say that the entangled electrons had the three opposite spin outcomes “builtin” for each axis in advance, but that of course leads to Bell’s inequality which conflicts with quantum mechanics. Because the spins are always opposite of each other whenever the two entangled electrons are measured on the same axis, Bell’s inequality provides a powerful tool for arguing against locality. I’ll construct a more rigorous argument for nonlocality a little later.
As a further illustration of why just rejecting deterministic hidden variables isn’t enough, consider the Copenhagen interpretation which denies hidden variable theory. The Copenhagen interpretation says that when the entangled electrons fly apart the electrons have no definite spin on any axis before any of the electrons have their spins measured, and so when both electrons are to be measured on the same axis, if electron #1 is measured first and found to be spin up (where the measurement randomly determines the spin result) this measurement influences the other electron’s measurement to be spin down when measured on the same axis. So the Copenhagen interpretation rejects hidden variables but embraces nonlocality. Indeed, it is difficult to see how to plausibly avoid some sort of nonlocality from taking place here.
What is Bell’s theorem?
Bell’s original theorem proved that a certain inequality follows from local hidden variable theory and showed how that inequality is inconsistent with quantum mechanics. “Bell’s inequality” now refers to a family of inequalities conceptually similar to the one used in Bell’s original theorem (all such inequalities assume locality and some form of hidden variable theory). These inequalities are also known as “inequalities of Bell’s type.” The term “Bell’s theorem” refers to a family of methods showing that (a type of) Bell’s inequality follows form (a form of) local hidden variable theory and that this inequality is inconsistent with quantum mechanics. Incidentally, it’s not just electrons and spin that exhibit this sort of entangled behavior. Similar behavior occurs with photons and polarization and one could use that to illustrate Bell’s theorem. In my version of Bell’s theorem, the inequality I derived from local hidden variable theory is this:
Pr(measured spins are different) ≥ 1⁄3Some forms of Bell’s theorem (like the one I used) show that given quantum mechanics, locality is incompatible with counterfactual definiteness. What is counterfactual definiteness? Counterfactual definiteness comes in varying degrees, but in its strongest version, it says that for any sufficiently specified conditions C, there are invariably true statements in the form of “If in (sufficiently specified) conditions C a measurement M was made, then R would have been the result.” Applied more narrowly to Bell’s theorem, counterfactual definiteness says that for any angle A and electron E, there are always true statements of the form “If we were to measure electron E at angle A, result R would have obtained.” An example of an interpretation that embraces this counterfactual definiteness is hidden variable theory largely thanks to its fully deterministic outlook, hence local hidden variable theory implying Bell’s inequality. An example of an interpretation that rejects it is the Copenhagen interpretation (it holds that observations randomly determine outcomes in some cases, such that “If we were to measure electron E at angle A, result R would have obtained” are not true, because different results are possible under those same circumstances). Bell’s theorem shows that counterfactual definiteness and locality cannot both be true if quantum mechanics is true thanks to Bell’s inequality. Though quantum mechanics doesn’t require counterfactual definiteness of the sort described, there is a (much weaker) sense of counterfactual definiteness that quantum mechanics does demand: whenever the entangled electrons have their first spin measurements along the same axis, the spins are always opposite of each other.
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