[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
(⟨ψ|⟨ψ|)(|ϕ⟩|ϕ⟩)=⟨ψ|⟨s|U†)(U|ϕ⟩|s⟩)=⟨ψ|ϕ⟩⟨s|s⟩=⟨ψ|ϕ⟩∴⟨ψ|ϕ⟩2=⟨ψ|ϕ⟩ - there are only two possibilities:
- ⟨ψ|ϕ⟩=1⇒|ψ⟩=|ϕ⟩
- ⟨ψ|ϕ⟩=0⇒|ψ⟩⊥|ϕ⟩
- 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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