[Quantum Information] No-cloning Theorem
No-cloning Theorem
Notes from RWTH Aachen University course
“Quantum Information” Summer semester 2020
professor: Müller, Markus
- A quantum computer cannot do
- There exists no phisical procedure to copy an arbitrary quantum state.
- proof:
- Assume that there are two state |ψ⟩,|ϕ⟩, and they can be copied by a unitary U
|ψ⟩|s⟩U→U|ψ⟩|s⟩=|ψ⟩|ψ⟩|ϕ⟩|s⟩U→U|ϕ⟩|s⟩=|ϕ⟩|ϕ⟩ - 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 - there are only two possibilities:
- \langle\psi|\phi\rangle = 1\Rightarrow |\psi\rangle=|\phi\rangle
- \langle\psi|\phi\rangle = 0\Rightarrow |\psi\rangle\perp|\phi\rangle
- We cannot copy arbitrary states.
Readings
W.K. Wootters and W.H. Zurek, A Single Quantum Cannot be Cloned, Nature 299 (1982), pp. 802–803
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