# How is the no-cloning theorem compatible with the fact that fan-out gates work?

I have some difficulty with understanding no-cloning theorem. Simply speaking, according to the theorem, it is not possible to copy a quantum state.

On the other hand, CNOT gate can be used as so-called fan-out gate which purpose is to copy one qubit to another one, previously in state $$|0\rangle$$.

It seems that these two facts negate each other.

My question: How is no-cloning theorem compatible with the fact that fan-out gate works?

By "copying a quantum state", we mean that we cannot take $$|\psi⟩|0⟩=\alpha|00⟩+\beta|10⟩$$ into $$|\psi⟩|\psi⟩=(\alpha|0⟩+\beta|1⟩)(\alpha|0⟩+\beta|1⟩)=\alpha^2|00⟩+\alpha\beta|01⟩+\beta\alpha|10⟩+\beta^2|11⟩$$ for arbitrary single qubit state $$|\psi⟩=\alpha|0⟩+\beta|1⟩$$. Notice that this resulting two-qubit state $$|\psi⟩|\psi⟩$$ is still separable.
But in the case of CNOT gate, it takes $$|\psi⟩|0⟩$$ to $$CNOT|\psi⟩|0⟩=CNOT(\alpha|00⟩+\beta|10⟩)=\alpha|00⟩+\beta|11⟩.$$ As you can see, the results are different. The resulting two-qubit state from CNOT is now entangled (not separable).