I've just made my changes.
In particular I:
* added some comments on complex numbers and bra-ket,
* made it explicit that state can be measured,
* explained/removed 'coherence' depending on its place in the text,
* made it more explicit that $p_1$ in the density matrix sum is NOT $|\alpha|^2$.
Unless I introduced more confusion (or grams, typos, ...) than cleared, this post should be ready to go.