I'd like to start by saying I agree with everything said above so far, |X| is not differentiable at X=0, but from the OP he's interested in a Quantum Mechanical situation. In QM we can (nay, we are compelled to based on our need for a rigged hilbert space as a basis) use generalized...
A mixed state is represented by a linear combination of any number of pure states, so to answer your question, no they are not the same.
My professor explained it like this:
A pure state (in classical or quantum mechanics) is as the name implies, pure. It contains all possible available...