# Quantum Entanglement

You have two entangled particles flying off in different directions. You interact with one to discover its spin and know its pair has the opposite spin. How is this different from having one black and one white poker chip sealed in envelopes. You mail one to a friend and when he opens his envelope he knows the color of your chip?

It’s different because in QM both poker chips are grey until you look at one. The colour is undefined - not definite but unknown.

This would be a Local Hidden Variable theory, a form of Local Realism. It has been ruled out by experimental proofs of Bell’s Theorem.

Bell’s Theorem shows that the results you get from observations of entangled particles are correlated, but cannot have been “preplanned” by carrying hidden data with the particles from the point of origin. The reasoning is subtle, but you can find numerous explanations that don’t require any math, e.g. this paper:
http://www.physics.wustl.edu/alford/physics/bell.pdf

Or many other briefer explanations/analogies to be found online, e.g. the one from Brian Greene’s book here:

I’ve always hated this answer, because it’s just a “because I said so”, and doesn’t address the reason why we say they’re unknowable, and so you could just as well say it about poker chips in envelopes.

First key point: You can’t measure a particle’s total spin. You can only measure one component of its spin, and see whether it’s positive or negative along that axis. So, for instance, you could measure one electron’s spin along the X axis, and find it to be positive. In that case, if you measure the other one’s spin along the X axis, you’re guaranteed to find that it’s negative. But once you measure one component of spin, you’ve changed the state of the system.

Second key point: You don’t have to measure both particles along the same axis. If you measure one particle along the X axis, and then measure the other one along the Y axis, the X measurement of the first will tell you absolutely nothing about the Y measurement of the second: It’s equally likely to be positive or negative. This is still consistent with a hidden-variables system; you just have to make your hidden variables slightly more sophisticated.

But the third key point: The angle between your measurement axes doesn’t have to be 90º (or a multiple thereof). You could, for instance, make one of your measurements along the X axis, and the other 45º between the X and Y axes. In this case, the results you get won’t match exactly, but will be correlated: The closer the two measurement axes are to each other, the greater the correlation. And even this is still consistent with hidden-variable models.

Finally, though, how much correlation is there? Bell was able to mathematically prove that, in any hidden-variable theory (with some other constraints which seem reasonable, like locality), there must be an upper bound for the correlation at any given angle. And at angles which aren’t a multiple of 90º, quantum mechanics predicts a greater correlation, outside of the bounds of Bell’s Inequality. Thus, quantum mechanics cannot be a local hidden variable theory. And the experiment has been done, with results as predicted by quantum mechanics, not as predicted by Bell’s Inequality.

Can you elaborate on the comment about locality? What exactly is locality? What happens to your reasoning if we don’t assume this? Is this the same kind of locality that (the absence of) is mentioned as a downside of the pilot wave interpretation?
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Locality means no instantaneous action at a distance (violation of SR). And yes, DeBroglie-Bohm (pilot wave) is a non-local hidden variable theory, the only kind of hidden variable theory that’s consistent with experimental results given Bell’s Theorem.

Understood. However I’m a bit confused as to how some other interpretation can claim to be local. If the spins are at an indeterminate state, and measuring one leads to an instantaneous effect on the range of possible states of the other, then how can this be considered as not involving instantaneous action?
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Nobody is willing to give up the impossibility of FTL signalling so the search for some way around invoking it continues. A hidden variable might exist that would explain it or something else nobody has yet found or postulated. The only thing everybody agrees upon is that no instantaneous anything happens.

What Is Real?: The Unfinished Quest for the Meaning of Quantum Physics by Adam Becker is a fascinating history of the various interpretations of QM. Each interpretation has a different way of attacking issues like these because the Copenhagen “shut up and calculate” mindset leaves questions that others feel itchy without having answers to. Becker takes sides in the controversy so a grain of salt is needed. Read it more for the questions than for answers.

That’s the whole point - there is no plausible non-local account. When Einstein et al wrote their paper, they asserted that quantum theory must be incomplete, because it involved paradoxical “spooky action at a distance”. Einstein coined this expression sardonically, as something he thought cannot be true. But, given Bell’s Theorem, all experimental results have since shown that EPR was wrong - the experiments do seem to imply spooky action at a distance. In other words, the data show that prior to observation the superposition is all there is to reality, there is no hidden underlying reality; only upon measurement of one particle does the other entangled particle “know” its know its correlated state; and it knows it instantaneously.

But it’s theories that are both Local and Realistic that are ruled out. This is usually interpreted as requiring a non-Local interpretation. I must say I’m quite unclear myself on what it could plausibly mean to retain Locality but instead violate Realism. Bell’s Theorem is based on correlations, there is no superluminal transfer of information. And inference from statistics depends upon the assumption that the observations are truly random, which is related to the notion of counterfactual definiteness and the no-conspiracy assumption. See here:

We can, in principle, explain the results by claiming that the statistical inference is invalid, see Superdeterminism.

But this amounts to asserting that the entire universe is like a vast conspiracy. I don’t think anyone takes it seriously. If it were true, all statistical inference would be invalid, so science wouldn’t work at all. (The John Bell quote in that article talks about “free will”, which I think is misleading. It’s nothing to do with free will, it’s a question of whether statistical inference is valid, which depends upon the randomization of measurements, not free will.)

Your confusion is understandable, the measurements in your example are always anticorrelated, which is perfectly consistent with a local classical explanation: i.e. the outcomes are already fixed. However Bell’s theorem, which uses a set-up where the measurements are not perfectly correlated, shows that, overall, the probabilities predicted by quantum theory do not match a classical local explanation.

ETA: I think Many Worlds would be considered Local, and violates conterfactual definiteness. I’m not really sure if that means it’s correct to say that it’s Local but not Realistic?

Many-worlds is both local and realistic, because it violates another, most often unstated, assumption of Bell’s reasoning: that experiment outcomes are single-valued, i. e. that you only either get a ‘spin-up’ or ‘spin-down’ result.

There’s indeed nothing special about discovering a property of one system, and immediately knowing the property of another—that’s just ordinary correlation. Entanglement, however, is correlation across different, and most importantly, mutually incompatible properties.

You know Heisenberg’s uncertainty relation: basically, it asserts that a quantum system can’t simultaneously have a definite position and momentum. Thus, position and momentum are incompatible properties. Another example of such incompatible properties is spin in x- and y-direction (equivalently, in the z-direction, but we only need two to illustrate the point). Thus, a quantum system can’t simultaneously have a definite x- and y-spin.

However, if you set a system of two particles up in the proper, entangled (Bell-) state, what you’ll find is that not only do they show perfect correlation along the x-axis, but likewise, that correlation exists along the y-axis, as well–despite the fact that neither of these can have both a definite x- and y-spin! That’s where the strangeness of entanglement comes from: if you have a single particle, and you measure its x- and then y-spin, then it’s easy to say that the particle just randomly decided on a particular value for the y-spin. But if that happens in entanglement, then if a particle is measured along the y-axis, and gives out a random answer made up on the spot, then in order to reproduce the fact that we will find its entangled partner having correlated y-spin, it must somehow tell its partner what it came up with—which would seem to necessitate some form of instantaneous communication.

This is what prompted Einstein, Podolsky and Rosen to set up their attack on quantum mechanics (the famous EPR-argument). They argued that the fact that since (counterfactually) if we measured the x-spin (they actually used position and momentum, with the spin-example being due to Bohm) of particle 1, we would immediately know the x-spin of particle 2, and if we measured the y-spin of particle 1, we would immediately know the y-spin of particle 2, there must be some pre-arranged agreement between the two particles. However, quantum mechanics knows nothing of this agreement: hence, they concluded, quantum mechanics must be incomplete.

That’s where Bell comes in (a little afterwards): he showed that every theory that includes such agreements (hidden variables), and furthermore, where there are no superluminal influences, must obey certain bounds, known as Bell inequalities. Quantum mechanics, however, violates these bounds (albeit not as much as it could, and why QM doesn’t violate these inequalities maximally is an interesting research programme these days). Consequently, either there must be some non-local influence, in order to reproduce the observed phenomena, or there can’t be any hidden variables.

Recently, people have brought up the option that the second requirement, actually, is moot: surely, there must be some recipe for coming up with measurement outcomes anyway; so one can’t just reject the assumption of ‘realism’ (hidden variables, what have you), or it’s not really an extra assumption.

I think everybody making this argument essentially sneaks in some extra assumption, about what a ‘reasonable’ theory should be like, which basically is equivalent to assuming a classical framework, but the assumption is somewhat technical to state. So giving up realism, to me, still is a live option. As for how such a theory would look like, well, just look at quantum mechanics: not every observable has a definite value at all times, and there are no non-local influences. Everything else, to me, seems to boil down to adding certain extra structure to QM.