This post is an explanation of a couple of points about entanglement that others get wrong. Part of the confusion is to mix two distinct paradoxes, so I separate them. (Remember a paradox is an apparent contradiction or confusing issue. Valid theories can have paradoxes.)

Uncertainty principle. A core tenet of quantum mechanics is that you cannot measure a particle's position and momentum at the same time. This is because particles are not really particles, and have wave-like properties that prevent having definite values for position and momentum.

Quantum mechanics enforces this uncertainty by using non-commuting observables. Measuring position then momentum is different from momentum then position. Other pairs of observables have this same property, such as Spin-X and Spin-Y.

This is an essential part of quantum mechanics, and was well-understood and non-controversial by about 1927.

Quantum twin paradox. If a system emits two equal and opposite particles, then properties of one can be deduced by measuring the other. For example, since momentum is conserved, the momentum of one will be opposite the other.

If the two particles are far apart, then knowledge about one seemingly has a spooky effect on our knowlegde about the other. This paradox occurs in either classical or quantum mechanics. It doesn't really violate the principle that there can be no action at a distance.

Combining these two paradoxes gives the EPR paradox. The idea is that you can measure the position of particle A and deduce the position of particle B, or you can measure the momentum of particle A and deduce the momentum of particle B, but you cannot measure the position and momentum at the same time.

Einstein argued in the 1935 EPR paper that this makes the theory of quantum mechanics incomplete. That is, you can deduce a particle's position and momentum by measuring its twin, but you cannot measure both at the same time. A complete theory would tell you both at the same time.

Bohm and Bell explain EPR with Spin-X and Spin-Y. You could use any noncommuting variables, as they all satisfy the uncertainty principle. Bohm proposed a nonlocal theory where a particle has a well-defined position and momentum all the time, but those variables might have nothing to do with what is observed. Bell proposed a classical theory of local hidden variables, but those theories have been refuted by experiments.

The EPR-Bohm-Bell followers will tell you that their argument is more subtle than just saying that the uncertainty principle makes quantum mechanics incomplete. That is because the position and momentum (or Spin-X and Spin-Y) are both predictable by measuring the twin particle. But you can't measure both at once in the twin particle, so you cannot predict both at once.

If you are bothered by the uncertainty principle, then you are going to be bothered by any theory were electrons have wave properties. Electrons are observed to have wave properties. If you are bothered by the quantum twin paradox, then you are also going to bothered by classical theories where someone might have info at a distance.

If you are not bothered by either paradox, then it is not clear why you would be bothered by the EPR paradox, because that is just putting the two paradoxes together. But there is a long list of intelligent physicists, from Einstein to Sean M. Carroll, who are tremendously confused by this combination.

Excellent analysis and explanation. ...Haven't run into any piece that simplifies the whole thing to this extent (but without dumbing it down). Congrats!

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--Ajit