Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 2 / 10D (Quantum Information and Computation)

For $|\psi ⟩=a|0⟩+b|1⟩$, the Controlled-NOT gate givesA cloned product would instead beEquality up to a physically irrelevant global phase requires $ab=0$. ThusFor a genuine superposition it creates quantum entanglement instead of two independent copies.

The no-cloning theorem for two pure states states that no single unitary operator can clone two distinct nonorthogonal states. Suppose, to the contrary, thatPreservation of the inner product by $U$ givesWriting $z=⟨{\alpha }_0|{\alpha }_1⟩$, this requires $z=0$ or $z=1$. The first case is orthogonality; the second means that the normalized states represent the same ray. Both contradict the assumptions. Therefore distinct nonorthogonal quantum states cannot be cloned by a unitary process.