This question is somehow the reverse of another question.
If a quantum system $S$ is in a pure state, then we can find a wavefunction that describes $S$. This wavefunction is unique up to a phase factor. We can find it from the density matrix.
This is not possible when $S$ is in mixed state. However, I expect that for every mixed state, $S$ is entangled to some system $T$ and the composite system $S+T$ can be described by a wavefunction.
For example: suppose that we prepare a spin system $S$ to be in the mixed state with probability 0.5 for spin-up and probability 0.5 for spin-down. Then we can not describe $S$ by a single wavefunction. But during the process of preparing $S$, we used some physical process $T$ to generate the probabilities of 0.5. Say that $T=|0\rangle$ would generate spin-up and $T=|1\rangle$ would generate spin-down. Then the composite system $S+T$ has the wavefunction $\psi = \frac12\sqrt2 |\uparrow\rangle \otimes |0\rangle+\frac12\sqrt2 |\downarrow\rangle \otimes |1\rangle$.
Is it possible to find such a $T$ for all mixed states $S$?
