Consider a Stern-Gerlach machine that measures the $z$-component of the spin of an electron. Suppose our electron's initial state is an equal superposition of
$$|\text{spin up}, \text{going right} \rangle, \quad |\text{spin down}, \text{going right} \rangle.$$
After going through the machine, the electron is deflected according to its spin, so we get
$$|\text{spin up}, \text{going up-right} \rangle, \quad |\text{spin down}, \text{going down-right} \rangle.$$
In a first quantum mechanics course, we say the spin has been measured. After all, if you trace out the momentum degree of freedom, we no longer have a spin superposition. In simpler words, you can figure out the spin by which way the electron is going.

In a second course, sometimes you hear this isn't _really_ a measurement: you can put the two beams through a second, upside-down Stern-Gerlach machine, to combine them into
$$|\text{spin up}, \text{going right} \rangle, \quad |\text{spin down}, \text{going right} \rangle.$$
Now the original spin superposition is restored, just as coherent as before. This point of view is advanced in [this lecture](https://www.youtube.com/watch?v=lZ3bPUKo5zc) and the [Feynman lectures](http://www.feynmanlectures.caltech.edu/III_05.html).

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Here's my problem with this argument. **Why doesn't the interaction change the state of the Stern-Gerlach machine?** I thought the two states would be
$$|\text{spin up}, \text{going up-right}, \text{SG down} \rangle, \quad |\text{spin down}, \text{going down-right}, \text{SG up} \rangle.$$
That is, if the machine pushes the electrons up, it itself must be pushed down by momentum conservation. After recombining the beams, the final states are
$$|\text{spin up}, \text{going right}, \text{SG down} \rangle, \quad |\text{spin down}, \text{going right}, \text{SG up} \rangle.$$
and the spins cannot interfere, because the Stern-Gerlach part of the state is different!

This is a special case of a general question: under what circumstances can interaction with a macroscopic system _not_ cause decoherence? Intuitively, there is always a backreaction from the spin onto the system, which changes its state and destroys coherence, so it seems that every particle is always continuously being measured.

In the case of a magnetic field acting on a spin, like in NMR, there is a resolution: the system state is a coherent state, because it's a macroscopic magnetic field, and coherent states are barely changed by $a$ or $a^\dagger$. But I don't think this argument applies for the Stern-Gerlach machine.