Quantum entanglement and spooky action at a distance

The assumption (if you cry it or not) "but the particle does have a definite spin, we just don't KNOW what it is, until it is measured! Duh!" is called realism, or in mathier speak, a theory of hidden variables.

Bell's inequalities now say that no theory that fulfills local realism (equivalently that has local hidden variables) can ever predict the correct results of a quantum mechanical experiment.

So we are faced with a problem: Do we give up locality or realism?

Most people choose realism, since giving up locality would totally destroy our conceptions of causality. It is possible that there is a non-local theory that assigns a definite value to every property at all times, but due to its non-locality, it would be even more unintuitive than "particles do not have definite properties".

There is no intuitive explanation for the non-realism of reality (there has to be a way to phrase that better...) because our intuitions have been forged in the macroscopic world which is, to good approximation, classical. But the non-realism is an effect that has no classical analogon, so we cannot understand it in pretty simple pictures or beautiful just-so stories.

Sometimes, we just have to take the world the way it is. (I have assumed that you do not want the whole QM story of non-commuting observables and eigenbases and so on to explain why we, formally from QM principles, expect realism to be false. If I have erred in that respect, just tell me)

At that instant, the other particle "assumes" the other spin.

This is where you go wrong, because things are actually weirder than that. I'll try to keep this simple, especially because if I don't I'd have to make extremely rigorous arguments given that this is a matter where different physicists may see things a bit differently.

A historical note : the view of QM that you now have was solidified with experiments that tested Bell inequalities. To put it shorty, these inequalities had to hold if QM was true, but if they did hold they definitively proved that "local realism" couldn't hold. "Local" (again to keep things simple) you can think of as meaning that the theory will never predict event B happening as a result of event A unless a signal moving at the speed of light has had the time to travel from A to B ("spooky action at a distance" violates this because it has to happen instantaneously). Realism you can think of as meaning your very objection to QM :

but the particle does have a definite spin, we just don't KNOW what it is, until it is measured!

Now here's the thing, laymen such as yourself might be tempted to give up on locality rather than realism. It is only in that case that you can even speak of "spooky action at a distance", because the fact is that if the particle doesn't have a definitive spin then no "spooky action" is required. If we accept that the probabilities predicted by QM are an intrinsic part of the very nature of the particle, then all we're seeing are the predicted correlations, not the result of one particle changing state because another was measured.

Giving up realism seems to bother you, so instead you may be tempted to say "Let's give up locality!". Unfortunately for you my friend, this does not work. QM you can think of as a sort of simplistic version of QFT, which is explicitly local in its first principles (at least for the Standard Model Lagrangian it is). Furthermore, it actually makes more sense to give up realism, if only because that's what the theory is actually telling you. The only reason we consider giving up locality is because a layman's intuition (or physicists before we finally just accepted that this was the nature of things) might be tempted to invent some convoluted non-local mechanism like "spooky action at a distance", but there's no reason to do this, and it just adds a needless and untestable layer of complications to your theory. In other words, it's actually far more natural to give up realism, because, again, that's what the theory is actually telling you.

I would just mention here that every probabilistic system, even classical, may exhibit a kind of "entanglement" or "spooky action at a distance" features.

For instance, imagine that you have $2$ boxes, and one bowl in each box. The bowl could have only a white or black color, and the two bowls have the same color. The boxes are closed, then one box stay on Earth, and the other box is sent to the planet March.

Imagine you are an external observer which do not put the bowls in the boxes, your analysis of the system is the following : one has the probability $\frac{1}{2}$ to find a white bowl in the two boxes, and a probability $\frac{1}{2}$ to find a black bowl in the two boxes. This is a probabilistic classical system.

Note this means, for you, that the bowls in the boxes have not a definite color.

Now you may open the box which stays on Earth. If you see a white bowl, you know immediately that the bowl in the box which is on March is white too, and you may imagine a "spooky action at a distance".

Of course, there is no "spooky action at a distance", and correlations are not causal relations. Here the correlations are about degrees of freedom (the color) which are completely independent of position degrees of freedom. So the correlations are exactly the same is the boxes are near each other, or very far from each other.

Quantum mechanics is a probabilistic theory, so much of the arguments above are correct for QM too. However, they are some specific features, because in QM, we are working with probablity amplitudes instead of probabilites, with vectorial states instead of "point" states, so the value of the correlations are very specific too, and you cannot always obtain these correlations even by a probabilistic classical system (Bell's theorem). So, quantum entanglement is special, sure, but in some sense, it is an extension of a probabilistic classical "entanglement".

Finally, I would say that it is far more interesting to compare probabilistic classical systems, and (probabilistic) quantum systems. It is not very interesting to compare determistic classical systems and QM.