[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 \(|\psi\rangle,|\phi\rangle\), 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 arbitrary states.

Readings

W.K. Wootters and W.H. Zurek, A Single Quantum Cannot be Cloned, Nature 299 (1982), pp. 802–803