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 Gelfand-Naimark-Segal 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 spaces1
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.