#### Re: "complex-valued probability distribution"

Of course you can never measure ""the value of the probability distribution".

The only way to plot is to prepare a large number of identical states, take a measurement, and mark the bin into which the measurement falls.

At the end of the day, you count the crosses in the bin, divide by the number of trials and then finagle a continuous function.

Just as in classical physics.

*What is a "state space"?*

Ok, let's use "sample space" for the individual classical outcomes. Sample space is just the possible values of the measurement with a funny notation:

For one qbit: |0> , |1>

For two qbit: |00>, |10> , |01> , |11> (the space gets larger quick - as 2^N)

For three qbit: |000>, |001>, |010>, |011>, |100> etc...

Now assign a complex-valued probability to each element of the sample space, under the constraint that the complex-valued vector of dimension 2^N has unit length (in the two-norm: sum of squared length of the individual components)

This is a straiightforward extension of classical probability, were you only assign real values to each element of the sample space , under the constraint that the real-valued vector of dimension 2^N has unit length (in the one-norm: sum of reals). It seems that this extension is the only one which makes mathematical AND physical sense.

"State space" of your current quantum state is the (Hilbert) space in which that complex-valued vector of 2^N dimensions "lives". This vector describes the full state of your N-qubit machine (it describes a "pure state"). There is the added finesses that shifting the "complex phase" of each component of the 2^N state space vector has no physical significance, so properly the actual state is actually a subspace of the state space, also called a "ray".

And that's about it. If you take a discrete space (of qubit arrays or anything else) and take it "to the limit" you get the continuous space where you need infinite-dimensional state space vectors - these may describe for example the position of a particle; every component of the vector giving the complex probability density function of "finding" a particle at the given position.