Successive measurment of incompatible variables on an entangled state

Question

Suppose I have 2 particles in an entangled state with opposite spins far away from each other, both stationary with respect to the lab. At time t=0 I measure the spin of both particles in the x direction. Suppose I got the results $S_1 = +1/2, S_2=-1/2$. After that, I preform a quick (simultaneous) measurement of the spin in the y direction, and shortly after another measurement of the spin in the x direction. Because I measured the spin in the y direction in between, I am no longer guaranteed to get the same results as before for the spin in the x direction. I have a 50% chance to get $S_1=-1/2, S_2=+1/2$. Suppose that is the case.

Now, if the particles are far enough from each other, and the time between the first and the last measurement is short enough, the spacetime interval $c^2\Delta t^2-\Delta x^2$ between the first measurement of particle A's x-spin to the second measurement of particle B's x-spin will be negative (i.e space-like). That means that there exists an inertial reference frame in which these two events are simultaneous. But, that means that in this frame both spins were measured to be $S_1=S_2=+1/2$ at the same time, contrary to the assumption that their spins are always opposite. How is it possible?

Answer 1

There is no contradiction. In any given frame, the quantum state of the two spins is different after a local measurement has been performed. The observer described in your second paragraph has the first measurement of particle B in his past light cone. If this observer believes that the spins should be anti-correlated, he is just ignorant. Once he finds that they are not anti-correlated, he has learned that something must have happened in his past to change the state of the spins (in this case a measurement on particle B).

Answer 2

Assuming no weak measurements ...

After either side does the first measurement, the pair are in an eigenstate of that measurement and if they were entangled in an anticorrelated state they are then in anticorrelated (but now unentangled) state after the first measurement. Thereafter all future measurements should proceed as normal for something in the state it is found to be in.

I'm actually not sure why you'd say they will be anticorrelated forever just because there was a time once upon a time when they were entangled.

I think it is easier to think of a spin measurement as a forced polarization. Before the "measurement" there were many possible states (three real degrees of freedom) then just a $\pm$ and a phase that barely matters. The point of entangled particles is that things you do here to affect them and that you do there to affect them both matter. Either location should be able to cause decoherence for instance, but now I'm edging towards weak measurements, so I'll stop.