# No-cloning theorem

- A quantum computer cannot do
- There exists no phisical procedure to copy an <ins>arbitrary</ins> quantum state.
- proof:
1. Assume that there are two state $|\psi\rangle,|\phi\rangle$, and they can be copied by a unitary $U$
$$|\psi\rangle|s\rangle\xrightarrow[]{U}U|\psi\rangle|s\rangle=|\psi\rangle|\psi\rangle \\
|\phi\rangle|s\rangle\xrightarrow[]{U}U|\phi\rangle|s\rangle=|\phi\rangle|\phi\rangle$$
2. take the inner product
$$(\langle\psi|\langle\psi|)(|\phi\rangle|\phi\rangle)=\langle\psi|\langle s|U^\dagger)(U|\phi\rangle|s\rangle)=\langle\psi|\phi\rangle\langle s|s\rangle=\langle\psi|\phi\rangle\\
\therefore \langle\psi|\phi\rangle^2=\langle\psi|\phi\rangle$$
3. there are only two possibilities:
1. $\langle\psi|\phi\rangle = 1\Rightarrow |\psi\rangle=|\phi\rangle$
2. $\langle\psi|\phi\rangle = 0\Rightarrow |\psi\rangle\perp|\phi\rangle$
4. *We cannot copy <ins>arbitrary</ins> states.*