(a) The classical controlled- $N O T$ operation applied to the 2-bit string $b 0$ (for $b=0$ or 1 ) achieves the cloning of $b$, i.e. the result is $b b$. Let $C X$ denote the quantum controlled$X$ (or controlled-NOT) operation on two qubits. For which qubit states $|\psi\rangle=a|0\rangle+b|1\rangle$ will the application of $C X$ to $|\psi\rangle|0\rangle$ (with the first qubit being the control qubit) achieve the cloning of $|\psi\rangle$ ? Justify your answer.

(b) Let $\left|\alpha_{0}\right\rangle$ and $\left|\alpha_{1}\right\rangle$ be two distinct non-orthogonal quantum states. State and prove the quantum no-cloning theorem for unitary processes.