What would happen if the opposite complementary variables of two entangled particles were measured at the same time?

Question

Many explanations about the uncertainty principle and its related EPR paradox state that it is impossible to measure opposite complementary variables on different entangled particles; for example, measuring the position of one particle and the momentum of the other at the exact time.

I fail to understand what it means for this to be "impossible" in this case. What would happen if such an attempt was made?

I can only think of 4 ways of interpreting this impossibility:

1. For every known pair of complementary variables, we only know how to measure one of the two variables for any given particle; for example, we only know how to measure position, not momentum (or vice versa). This is obviously incorrect.
2. It is impossible to measure any of the complementary variables of a particle at the exact time we measure any other complementary variable of another particle. For example, given any two particles, we cannot measure both of their positions at the same time. I'm guessing this is also incorrect.
3. Same as #2, but exclusive for entangled particles, implying that there is something in the process of the creation of such a pair that prevents them from being measured at the same exact time.
4. In the case of an entangled pair being measured at the same time for opposite complementary variables, these measurements would "somehow" no longer correlate between the two, thus rendering the particles entangled no more.

Furthermore, if #4 is the case, was it verified? Was there an experiment showing the measured particles indeed were no longer entangled, perhaps by proof of contradiction; i.e, observing a later measurement that could only occur if these values were different than the values assumed based on the original measurements and the correlation between the particles?

I would love it if the answer would be simple as possible, as I do not have a strong background in neither physics nor mathematics.

Answer 1

This has really been fully answered in the comments, but here is more detail:

You want the particles to be in an entangled state that guarantees that momentum measurements on both particles are certain to agree. So take the entangled state $\xi=A\otimes A+B\otimes B$, where $A$ and $B$ are eigenstates of momentum. ("Momentum" here can stand for any observable of your choice.)

Then $X=A+B$ and $Y=A-B$ are eigenstates of the complementary observable (which we can call "position"). Check that (up to multiplication by a constant)

$$\xi=X\otimes X+Y\otimes Y$$

so that position measurements of both particles are also certain to agree.

But

$$\xi=X\otimes A+X\otimes B+Y\otimes A-Y\otimes B$$ so that if you measure the position of the first particle and the momentum of the second, all four outcomes are equally likely.

Moreover, if you measure the position of the first particle and it comes out to be, say, $Y$, then the state collapses to $Y\otimes A-Y\otimes B$, leaving you with maximum uncertainty about the outcome of a momentum measurement on the second.

Answer 2

If you measure the position of one of two entangles particles with infinite precision this doesn't mean that you automatically know the position of the other particle with infinite precision (even though they are entangled). Likewise for momentum measurement. Position and momentum are not quantum entangled. Position and momentum entanglement hold only for the individual particles. If you measure the position of one particle, you know instantaneously that is not at some other position. Likewise for momentum. For spin, on the contrary, you know instantaneously that the spin of the other is opposite if you measure it to be up or down (in the case of fermions).