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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?

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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).

CNOT gate does not copy qubit states; rather it creates entanglement between them.

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  • $\begingroup$ It is perhaps worth adding that controlled not does perform fanout if your input is in the computational basis, but only if it is in the computational basis. $\endgroup$ – DaftWullie Dec 9 '19 at 6:18

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