No-cloning theorem

It is impossible to create an independent copy of a given Qubit in an unknown state $|\psi ⟩$.

Proof

All actions on a Quantum Hilbert Space can be represented by unitary operators. The map $|\psi ⟩\times |\varphi ⟩\to |\psi ⟩\times |\psi ⟩$, mapping $H^2\mapsto H^2$ is not unitary. Hence, there is no transformation that clones a Qubit.