quantum mechanics - EPR paradox and uncertainty principle

In Wikipedia article EPR paradox,

The original paper purports to describe what must happen to "two systems I and II, which we permit to interact ...", and, after some time, "we suppose that there is no longer any interaction between the two parts." In the words of Kumar (2009), the EPR description involves "two particles, A and B, [which] interact briefly and then move off in opposite directions."[9] According to Heisenberg's uncertainty principle, it is impossible to measure both the momentum and the position of particle B exactly. However, according to Kumar, it is possible to measure the exact position of particle A. By calculation, therefore, with the exact position of particle A known, the exact position of particle B can be known. Also, the exact momentum of particle B can be measured, so the exact momentum of particle A can be worked out. Kumar writes: "EPR argued that they had proved that ... [particle] B can have simultaneously exact values of position and momentum. ... Particle B has a position that is real and a momentum that is real."

But isn't measurement in quantum mechanics not related to Heisenberg uncertainty principle? According to my knowledge, measurement collapses wavefunction into one basis state, and has nothing to do with uncertainty principle..

I am bit confused of the paper.

Answer

The idea is as follows:

Suppose we create (for example) a pair of electrons that fly off in opposite directions. Because of conservation of momentum we know the momenta must be equal and opposite so if we measure the momentum of one of the particles we know the momentum of the other particle. Likewise if we measure the position of one of the particles we can calculate the position of the other.

So we wait until the particles are a long way apart like a light-year (remember this is just a thought experiment!) and we measure the momentum of particle A to perfect accuracy (so it's position is unknown) and the position of particle B perfectly (so it's momentum is unknown). So we know the momentum of particle A perfectly, but from our measurement of B's position we also calculate the position of particle A perfectly. The end result is that we know both the momentum of particle A and it's position i.e.

$$ \Delta x_A \Delta p_A = 0 $$

and this contradicts the Heisenberg uncertainty principle.

Note that I said the two particles are a light-year apart. We specified this so any signal from particle A to B, or vice verse, would take at least a year to travel. This means as long as we do our measurements of A and B within a year the two measurements can't affect each other (because information can't travel faster than light).

The resolution of the paradox is that anything you do to an entangled system affects the whole system simultaneously i.e. there is no limitation due to the speed of light. You can't just do a measurement of particle A without affecting particle B even when they are many light years apart. Any measurement you do is necessarily a measurement on the whole entangled system and you will always find that Heisenberg's uncertainty principle applies i.e.

$$ \Delta x_{AB} \Delta p_{AB} \ge \frac{\hbar}{2} $$

This is unpalatable to lots of people because it suggests that quantum entanglement is necessarily non-local. Lots of effort has been expended in trying to get round this e.g. hidden variable theories. However the evidence to date is that entanglement really is non-local.