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On page $402$ of Nielsen and Chuang, there is the statement: "there is no notion in quantum mechanics analogous to the joint probability distribution for variables X and Y that exist at different times".
Please help me understand this.
Said differently, if classically I can analyze the joint distribution between two variables X and Y that exist at different times, why can't I do the same when X and Y are quantum variables?
The gist is that to measure the joint distribution you need "numbers" associated to one random variables and numbers associated to the other one. But in QM, to "know something about" $X$ at time $t$ means you measured the state at that time, and therefore at least partially collapsed the state, and therefore changed what you'll observed measuring $Y$ at some future time $t+\Delta t$.
You'll still be able to get correlations between the two measurement processes, provided the intermediate measure doesn't completely destroy the state. But the resulting correlations will be a rather different beast than what they are classically. In particular, the probability distribution on $Y$ will depend on the observed outcome for $X$, and on the fact that you measured it at all, for that matter.
Consider as a simple example measuring in the $Z$ basis at some time, then applying the identity evolution (that is, "evolve" with the trivial evolution which changes nothing), and then measuring $X$ afterwards. Assume the initial state is $|+\rangle$. If you don't measure $Z$, the outcome distribution when you measure $X$ will be the deterministic $(1,0)$. If you instead measured $Z$ and found whatever result, the probability distribution associated to measuring $X$ will be $(1/2,1/2)$. You can still define a joint probability distribution in such cases. Here you'd get $p_{0i}=1/2$ and $p_{1i}=0$. But note the difference in interpretation compared to the classical situation: here you're not measuring correlations between random variables which exist independently on the way you're measuring them; rather, you're measuring correlations between specific choices of measurement outcomes at different times.
