add DensityMatrix.to_statevector() for pure states

[ajavadia]

This is handy when you have a circuit in a product state and you want the reduced state as a statevector (e.g. discarding an uncomputed ancilla).

[chriseclectic]

This function should take an atol value for comparing small eigenvalues close to zero. You could also check that the current density matrix is hermitian before computing the eigen decomposition since you can fail early if its not (DensityMatrix objects could be in an "invalid" quantum state before calling this function).

After this as Jay says you should check the number of eigenvalues rather than the purity. I would check that there is only 1 non-zero eigenvalue up to the absolute tolerance. Something like this:

def to_statevector(self, atol=None, rtol=None):
    if atol is None:
            atol = self.atol
    if rtol is None:
            rtol = self.rtol

    if not is_hermitian_matrix(self._data, atol=atol, rtol=rtol):
        raise QiskitError("Not a valid density matrix")

    evals, evecs = np.linalg.eigh(self._data)
    
    if len(evals[abs(evals) > atol]) != 1:
        raise QiskitError("Density matrix is not a pure state")

    return Statevector(np.sqrt(np.max(evals)) * evals[:, np.argmax(evals)])

The last line allows for the case where the density matrix is a rank-1 projector, but not normalized to trace 1. It can still be converted into a vector, but that vector will not be normalized.

[ajavadia]

@chriseclectic @jaygambetta addressed your comments.
Chris I restricted it to trace-1 density matrices for now as that is the is_valid definition currently.