[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:


    1. Assume that there are two state |ψ,|ϕ, and they can be copied by a unitary U
      |ψ|sUU|ψ|s=|ψ|ψ|ϕ|sUU|ϕ|s=|ϕ|ϕ
    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 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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