Stern-Gerlach experiment and probability theory

Question

I'm trying to understand why, precisely, we cannot use classical probability theory in quantum mechanics. I came across an explanation of the Stern-Gerlach experiment, saying that if $X$ and $Z$ are Bernoulli random variables for the $x$ and $z$-spin of an electron, if we measure the $z$ spin first, the expectation of the $x$ spin is always $0$. If we measure the $x$ spin first, the expectation of the $z$ spin is always $0$. The idea is that this then contradicts $\mathbb{E}(XZ)=\mathbb{E}(ZX)$ from classical probability theory. I have two questions.

Why does the random variable $XZ$ represent a measurement of $X$ first, and then $Z$? I would think the conditional expectations $\mathbb{E}(X|z)$ and $\mathbb{E}(Z|x)$ are the relevant things to look at. But then what does this have to do with non-commutativity of $X$ and $Z$?

I think I'm also not exactly understanding the claims of the experiment as well. My understanding is, you need three Stern-Gerlach detectors to see something interesting. You send an electron through a $Z$ field and filter out the down spins, then you send the result through an $X$ field, and filter out the down $x$-spins, and you send the result through another $Z$ field and you get a 50-50 distribution of $z$-spins. I'm not exacting seeing how this is the same as the explanation in the first paragraph. If the spins are +1/2 and -1/2, the expectation of $X$ and $Z$ should be $0$.

Answer 1

Not sure what sources you used so far, but there is a limit to how much you can expect from the forum. There are good books on quantum mechanics out there. For your types of questions, I would probably suggest something like Ballentine

The difference between how, I think, you are thinking about $\mathbb{E}\left[J_X\right]$ and how one would think about it in QM, is that the latter is very much about changing the state of the system, you subject system to measurement, it's state collapses into one of the possible states, this gives you the measurement, following this the system evolves from the state it collapsed into. Hence if you measure $J_X$ and then $J_Z$ one can talk about order of these, and it is not the same thing as conditioning joint distribution to extract $\mathbb{E}\left[J_X|j_z\right]$. I think it is more like $do$-calculus operations in a causal model.

It so happens that linear operators, with their commutation relations, can encode the inherent order one imposes on the system when one measures one property and then another, collapsing the system state after each measurement.