Singlet states and the estimation of eigenstates and eigenvalues of an unknown Controlled-U gate

We consider several problems that involve finding the eigenvalues and generating the eigenstates of unknown unitary gates. We first examine Controlled-U gates that act on qubits, and assume that we know the eigenvalues. It is then shown how to use singlet states to produce qubits in the eigenstates of the gate. We then remove the assumption that we know the eigenvalues and show how to both find the eigenvalues and produce qubits in the eigenstates. Finally, we look at the case where the unitary operator acts on qutrits and has eigenvalues of 1 and -1, where the eigenvalue 1 is doubly degenerate. The eigenstates are unknown. We are able to use a singlet state to produce a qutrit in the eigenstate corresponding to the -1 eigenvalue.