No-cloning Theorem

can copy a bit value in the first qubit:

when

But can it copy a whole superposition state?

NO!!!

Theorem

Let be any set of states of a quantum system, that contains at least one pair of non orthogonal states.
Then there is no unitary cloning process that achieves cloning for all states in .

Proof

By contradiction.
Assume there is a universal quantum cloner
Suppose and such that and
Quantum cloner needs to do

Since operator is unitary

therefore