I've been dealing with quantum mechanics for a while now and most of the time i feel like i'm pretty comfortable with the basics. I know wave-functions are fundamentally different from particles even though they sometimes get used interchangeably. I'm okay with bras and kets and using operators for things like position and momentum. But every once in a while something reminds me that i'm still glossing over a lot of the weirdness in my head.
There's a famous argument between Einstein and Bohr about wave-functions. Basically, Einstein argues that the Heisenberg Uncertainty Principle doesn't mean that particles don't have definite positions and momenta, just that we can't measure them. (Search "Einstein-Podolsky-Rosen" or "EPR Paradox") The consensus of most of the physics community is that Bohr successfully defended the indefinite nature of quantum mechanics through a series of thought experiments. (Einstein's assumptions are referred to as 'local reality', giving Bohr's alternative the unhelpful moniker 'non-reality'.) Thought experiments are all well and good if you already have your Nobel Prize, but a Prof. John Bell of CERN actually proposed a class of experiments to distinguish between a fundamentally classical world and a fundamentally quantum one. Many of the experiments are quite subtle (and therefore easy for the layman to ignore), but one of them really bothers me.
Suppose you have a black box filled with spin-0 particles which occasionally decay into a pair of spin-1/2 particles. Each pair will necessarily go in opposite directions (conservation of momentum) and have opposite spin (up or down, conservation of angular momentum). Until they reach some outside observer, they are 'entangled' because the result of one spin measurement will depend on the result of the other. This type of system has been constructed and indeed if a pair of observers agree what axis they are using before-hand they always get opposite results. If Observer 2 rotates his axis by 90 degrees, his measurements are now completely uncorrelated to Observer 1's. But that's okay; if you had two random vectors pointing in opposite directions, knowing the x-component of one would tell you nothing about the y-component of the other.
Where things get weird is when you let the measurement axes float around. Suppose you do the same experiment, but for each decay Observer 1 uses a vertical measurement axis while Observer 2 randomly selects an axis which is either vertical or rotated by x or by 2x. Later they compare their data, only then revealing the relative angle between their measurements. If the angles are aligned, they get 100% correlation. If misaligned slightly, the correlation is still large but not 100%. If you imagine the spins as classical vectors whose orientation is merely unknown (all you can measure is 'up from horizontal' or 'down from horizontal') then the chance of a rotation by x degrees pushing the result through horizontal is x/180. The chance for a rotation of 2x is 2x/180. So if the results of measurements skewed by x are correlated 1-a of the time then measurements skewed by 2x will be correlated 1-2a of the time provided x is smaller than 45 degrees. But if the spins are quantum spins, then you need to calculate the wave-function overlap, which works out to cos(x). For small x, this makes the correlation for measurements skewed by x equal to 1-(1/2)*x^2 == 1-a, but for a skew of 2x, it is 1-(1/2)*(2x)^2 = 1-4a ! Doubling the angle decreases the correlation 4-fold.
To get how crazy this is, imagine you had a pair of very weighted coins which flipped heads 99% of the time. If you flip one of them a bunch of times, the results are 99% correlated to 'all heads'. Now you flip the pair of them a bunch of times and record how often they show the same side. You would expect the correlation to be at least 98% since the combined incidence of 'not heads' is only 2%. But if you're flipping quantum mechanical coins, you could get only 96% correlation. They don't match each other more often than they collectively don't match some third reference flip. This is a sign that they don't have a definite direction, even a hidden one, until you look at them. Only the correlation can be measured.
The ideal of 'realism' (that uncertainty doesn't prevent unknowable variables from existing) is closely linked to the idea of 'counterfactual definiteness', the idea that an experiment you didn't perform still has a definite result. In the case of the coins, this would mean checking that either coin matches 'heads' rather than checking that they match each other. For quantum spins, counterfactual definiteness says that even though you didn't measure the entangled states along parallel axes, they would have had opposite spins if you had. Non-reality says that question is meaningless, even though it can be answered in the abstract with no uncertainty.
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