Will the direction of two entangles particles going through another set of Stern—Gerlach magnets after an initial set, be opposite?

Question

Consider this experimental set-up:

• You have a source emitting two entangled 1/2-spin particles, one left, one right.
• You have two 180-degree-oppositely, horizontally oriented Stern—Gerlach magnets, one on the left, one on the right.
  • The particles have opposite spin, and the magnets are oppositely oriented, so the particles will either both go 'up' or both go 'down', 100% of the time, right?
• We ignore, for now the particles that went 'down'.
• At this point we should have two up-particles fully oriented with the horizontal Stern—Gerlach magnets, is that right? So if we were to put them both through the same magnets again, they would both go 'up' again?
• Now say that after this point we put them through a set of Stern—Gerlach magnets that are 45-degree oriented clockwise from the initial set. So one has a net 45 degree rotation, the other has a net 225 degree rotation (but a relative 45 degrees from its initial orientation). Given the particles are perfectly aligned, this gives an ~85.36% chance to go 'up' (cos^2 (45/2)) along the nearer trajectory and ~14.64% chance to go 'down' along the further trajectory.

My question is: will both particles always go in the 'same' direction, e.g. both take the 85.36% path or both take the 14.64% path? Or will they take opposite paths? Or are they no longer correlated with how they will go from here / will they go independently?

Answer 1

Let's say your particles start in the state $\frac 1 {\sqrt 2} \left(|ud\rangle + |du \rangle\right)$. In your example, particles in this entangled state are deflected by magnets twice, where the magnets are pointed in opposite directions. If we were to put a screen behind the first set of magnets we would always see either both particles go up, or both down, just as you said.

Let's say the first magnet directs both particles in the $|ud\rangle$ state up. Then if we add additional magnets that act only on particles deflected up, these magnets will only act on $|ud\rangle$, and any particles reaching the later magnets effectively been prepared in the $|ud\rangle$ state! This doesn't have any effect on the next set of magnets which, as you describe, will deflect both particles up again. But when the particles get to the third magnet, the correlation between the spins no longer exists, and we are no longer guaranteed to get the same outcome.