Quote by Fra
Vanesh, in your view... do you ever attempt to explain where the complex amplitude formalism comes from, or is it just taken to be a fact, and you are trying to interpret it?

For the record...
If, given a quantum state and a measurement operator, you have some means of extracting the "expected value" of the operator... then the GelfandNaimarkSegal construction says that you can take the 'square root' of the state, giving you a bra and a ket that represent your quantum state. Such objects live in Hilbert spaces.
This applies to any state  including statistical mixtures. In the case of a statistical mixtrue, the Hilbert spaces
^{1} produced by the GNS construction has a special form; it is
reducible. You can split the Hilbert space into irreducible state spaces (e.g. you can split "particle with unit charge" into "particle with charge +1" and "particle with charge 1"). The state corresponding to a statistical mixture can always be decomposed into its individual parts.
The same cannot be said for pure states; the GNS construction provides you with an irreducible Hilbert space. Thus we see that pure states
cannot be reinterpreted as statistical mixtures. (At least, not in any direct way)
1: more precisely, it's a unitary representation of the measurement algebra.