But those have real valued probability distributions too, right? Does modeling the evolution of the joint distribution of all the observables not count?
There is not really a meaningful joint distribution. Some quantum observables are incompatible with each other which essentially means they can't be assigned simultaneous values.
More precisely a quantum observable is a map (a function) that takes in a quantum state and outputs a probability distribution, representing the probabilities of the various outcomes you could get if you measure that observable on that state. The equivalent statement is also true of classical observables and classical states.
Under classical rules it turns out that if you have many observables acting on the same system you can come up with a joint observable, that maps a state to a joint probability distribution for all the observables. For incompatible quantum observables this is emphatically not the case. Given two quantum observables there is generally not a joint observable representing simultaneous measurement of them.
Yes you're right, every superposition of quantum systems have a real valued probability distribution for some observable variable.
QM has very specific rules when it comes to how two random variables are dependent. That specific rule comes up when you have quantum interference and this is best described by the conjugate square of two complex functions A^2 + B^2 + A*B + B*A. This is just a special case of the union probability: P(A) + P(B) - P(AB).
Yes, but you have more variables than necessary. Why model all the observables independently when you can model the system with one complex wave function?
It's the interaction between variables that is key.
More precisely a quantum observable is a map (a function) that takes in a quantum state and outputs a probability distribution, representing the probabilities of the various outcomes you could get if you measure that observable on that state. The equivalent statement is also true of classical observables and classical states.
Under classical rules it turns out that if you have many observables acting on the same system you can come up with a joint observable, that maps a state to a joint probability distribution for all the observables. For incompatible quantum observables this is emphatically not the case. Given two quantum observables there is generally not a joint observable representing simultaneous measurement of them.