Premise of the proof of the No-Cloning Theorem

Question

I have seen two similar proofs of the no-cloning theorem. They assume (to the contrary) that there exists a unitary operator $U$ such that $U |\psi\rangle |0 \rangle = | \psi \rangle | \psi \rangle$, For any possible $|\psi\rangle$. The proof does not seem to rule out the case that there exists a specific $U$ that can clone only the specific state $| \psi \rangle$. Discussion of the no-cloning theorem implies that there cannot be a specific $U$, which can only clone a certain state, even when the proof only proves that there cannot be a general $U$ which can clone any state. Is there a proof of this specific case somewhere? Or maybe I am missing something from the original proof.

(I am referencing the one in Nielsen and Chuang which ends with the contradicition that $\langle \psi | \phi \rangle = \langle \psi | \phi \rangle^2$.)

Answer 1

The proof does not seem to rule out the case that there exists a specific U that can clone only the specific state |ψ⟩.

That's because you can clone specific states. Cloning is only impossible if the set of possible input states includes a pair of states that are not orthogonal.

For example, here is a circuit that performs $|\psi⟩ \to |\psi⟩|\psi⟩$ as long as $|\psi\rangle$ is promised to be exactly $|0\rangle$ or exactly $|1\rangle$ and never anything else:

Answer 2

The real missing keyword in the stating the theorem is "arbitrary unknown state"! If you have some information about $|\psi\rangle$, i.e. specific state, then perhaps you can reconstruct that state!