Quantum entanglement and uncertainty

Question

I have a question about measuring entangled particles and the uncertainty principle. I know that this has been asked before, but I am still not clear on the explanations, so I will try to explain why I am confused.

Assumption: If two particles are entangled, then by measuring the position of one, we know what would be the result of measuring the position of the other. Likewise for momentum.

Say we have two particles A and B that are entangled. We measure the position of A and thus know A’s position. We measure the momentum of B and thus know B’s momentum. From my assumption, it would seem to follow that we also know what A’s position would have been had we measured it. Why is it unreasonable to infer that momentum A was equal to momentum B at the time (or just before the time) when we measured position A?

I want to present two explanations I have seen and my problems with them:

Explanation 1: After measuring position A, momentum A and momentum B become uncertain.

Issue: What does it mean for a momentum to become uncertain? I suspect that my confusion may lie here. Obviously I can still measure momentum B and obtain a result. Is this result somehow not accurate? If so, what experiment can be done to prove that it is not accurate?

Explanation 2: After measuring position A, momentum A and momentum B no longer give the same result when measured

Issue: This could be the same as saying that measuring position A changes momentum A. Still, I do not see why it is unreasonable to infer that momentum A was equal to momentum B at the time (or before the time) that position A was measured. Does my choice of measurement on A affect the value I measure for particle B? That is, does quantum mechanics imply the possibility of measuring two distinct values for momentum B, depending on whether or not position A has been measured? If so, can this be demonstrated by experiment?

Answer 1

Explanation 1: After measuring position A, momentum A and momentum B become uncertain.

Issue: What does it mean for a momentum to become uncertain? I suspect that my confusion may lie here. Obviously I can still measure momentum B and obtain a result. Is this result somehow not accurate? If so, what experiment can be done to prove that it is not accurate?

Explanation 2: After measuring position A, momentum A and momentum B no longer give the same result when measured

By measuring position A, you will know B's position, and at the same time you would have broken their entanglement. Meaning that if you now measure either of the momenta, they will be uncorrelated, and so measuring B won't tell you about A

Still, I do not see why it is unreasonable to infer that momentum A was equal to momentum B at the time (or before the time) that position A was measured.

The problem is that before we measure their momentum, they don't have a definite momentum, so you can't talk about A's or B's momentum, being equal or anything, they are in a superposition.

Answer 2

Explanation 3:

How were the entangled particles created?

Suppose they started as one particle, which then split into two. This original particle must have had some uncertainty in its position and momentum. Let's assume that there was little uncertainty in the original particle's position. Then there must have been lots of uncertainty in its momentum. The sum of the momenta of A and B is the original particle's momentum, so even if we know A's momentum exactly, we can only infer B's momentum to limited accuracy.

Answer 3

Uncertainty principle has to be valid for both particles A and B individually. But as long as the entanglement sustain between them then it also valid for mutual combination.