Erratum in Schlosshauer's "Decoherence" textbook (on the outcome of a SG experiment)? In his textbook on Decoherence, Schlosshauer is trying to explicate the difference between a superposition and a classical ensemble. Considering a two-level system and using fairly standard notation, Schlosshauer writes that for a pure state $|0_z\rangle = \frac{1}{\sqrt{2}}(|0_x\rangle+|1_x\rangle)$ we would find that a measurement of $\sigma_z$ in an SG apparatus would obtain a spot "on the upper region of the screen" (0 corresponding to spin up). Schlossahuer then writes:

If the superposition did represent a classical ensemble of the states $|0_x⟩$ and $|1_x⟩$, i.e., if each electron actually was in either of the two states, a single spot in the center of the screen would appear, since the inhomogeneity of the magnetic
field is oriented along the z axis only and would therefore not induce any
splitting of the beam of atoms.

Is this correct? I would have thought based on my understanding of this experiment from e.g. Sakurai that we would obtain a 50-50 distribution of up and down. Indeed, if I take the corresponding (incoherent) mixture to which Schlosshauer refers as $\rho = \frac{1}{2}(|0_x\rangle\langle 0_x|+|1_x\rangle\langle 1_x|)$ then
$$\textrm{Prob(up in z direction) = Tr}(\rho |0_z\rangle\langle 0_z|) = \frac{1}{2}$$
What am I missing here? Schlosshauer seems to be ascribing some modicum of classicality to the spin state saying that "since it's in the x-direction" it "won't be affected in the z-direction".
I have also attached the full discussion in case I am truly erring somewhere in understanding what he means. Surely I am making a mistake somewhere as this seems to be an overtly simple error for a physicist like Schlosshauer.



 A: I don't think you're missing anything. The problem is simply that "classical ensemble" in this context could have several meanings and the author got confused about them.
As you say

Schlosshauer seems to be ascribing some modicum of classicality to the spin state saying that "since it's in the x-direction" it "won't be affected in the z-direction".

that is, the claim is correct if we treat the idea of the $\lvert 0_x\rangle$ as classical states where the spin vector is purely in the $x$-direction. This, of course, doesn't really have anything to do with the way in which we use "classical" when we talk about a density matrix/incoherent mixture of classical states as a classical ensemble.
You are equally correct that the incoherent mixture of $\lvert 0_x\rangle$ and $\lvert 1_x\rangle$ would predict a 50:50 split of the beam.


Surely I am making a mistake somewhere as this seems to be an overtly simple error for a physicist like Schlosshauer.

This doesn't really change that the text is incorrect but this mindset - that experts should not be expected to make simple errors - misunderstands the nature of expertise. In fact, it is exactly simple errors without consequence that experts often make. The expert already knows the correct answer, and a simple mistake in the argument that doesn't change the conclusion will only be caught when someone thinks through each step of the argument carefully. But the expert has thought about this conclusion a hundred times and knows it's correct, so the diligence with which experts review this kind of argument is often subconsciously low; they know their conclusion is correct, after all.
In this case, the author knew that the Stern-Gerlach experiment can be used to argue that quantum superpositions are not classical statistical mixtures and he wrote down an argument for it. The argument is wrong, but it is wrong in a way - mistaking one notion of "classical" for the other - that is easy to make on the fly and this error doesn't actually affect anything else. We've established a correct statement - quantum superpositions are not statistical mixtures - after all, nothing we derive from that after this is going to be negatively affected by our argument for the truth of this statement being wrong.
Of course, there are instances where such simple errors may not be benign and actually do lead to wrong conclusions; this is not an argument not to check every step of every argument you read carefully. I'm just saying that the rate with which experts make careless mistakes is usually not lower than the rate with which other people make mistakes - but the expert can catch the errors that lead to disastrous consequences more confidently than the novice. Likewise, experts can tend to ignore errors without consequence because they know how the argument is "supposed to go".
