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- State passing through Stern-Gerlach magnet w/o projective measurement. How to treat it?

Suppose we have a state ##|\psi\rangle \sim |\uparrow z\rangle + |\downarrow z\rangle## passing through a Stern-Gerlach magnetic field oriented in the ##\hat{z}## direction such that ##|\uparrow z\rangle##'s are pulled up and ##|\downarrow z\rangle##'s are pulled down. Then suppose we place some operation ##M_\uparrow## on the top path and ##M_\downarrow## on the bottom path and both paths are later recombined.

If we treated the whole set-up as some kind of black-box where we don't obtain read-outs for ##M_{\uparrow/\downarrow}##, what state would we observe post-recombination?

Would it be ##|\psi\rangle \sim M_\uparrow|\uparrow z\rangle + M_\downarrow|\downarrow z\rangle##? Or even ##|\psi\rangle \sim M_\uparrow|\uparrow z\rangle|\text{u}\rangle + M_\downarrow|\downarrow z\rangle|\text{d}\rangle##, where ##u,d## represent "path degrees-of-freedom".

My guess is that it's the former since ##|\uparrow z\rangle## "intrinsically" specifies the upwards path in such a setup. But I don't really have anything concrete to check this guess against. Assistance would be greatly appreciated!

If we treated the whole set-up as some kind of black-box where we don't obtain read-outs for ##M_{\uparrow/\downarrow}##, what state would we observe post-recombination?

Would it be ##|\psi\rangle \sim M_\uparrow|\uparrow z\rangle + M_\downarrow|\downarrow z\rangle##? Or even ##|\psi\rangle \sim M_\uparrow|\uparrow z\rangle|\text{u}\rangle + M_\downarrow|\downarrow z\rangle|\text{d}\rangle##, where ##u,d## represent "path degrees-of-freedom".

My guess is that it's the former since ##|\uparrow z\rangle## "intrinsically" specifies the upwards path in such a setup. But I don't really have anything concrete to check this guess against. Assistance would be greatly appreciated!