Quantum teleportation

Protocol

1. The Bell state shared by \(A\) and \(B\) is available

   • \(A:1/\sqrt{2}(|0_A0_B\rangle+|1_A1_B\rangle)\)
   • \(B:1/\sqrt{2}(|0_A0_B\rangle+|1_A1_B\rangle)\)

2. \(A\) interacts her qubit with the (unknown) state \(|\psi\rangle\)

   • assume that \(|\psi\rangle = \alpha|0\rangle+\beta|1\rangle\)

   \(\begin{aligned}A:&(\alpha|0\rangle+\beta|1\rangle)(1/\sqrt{2}(|0_A0_B\rangle+|1_A1_B\rangle))\\ \xrightarrow[]{C_1NOT_A}&1/\sqrt{2}[\alpha|0\rangle (|0_A0_B\rangle+|1_A1_B\rangle)+\beta|1\rangle(|0_A1_B\rangle+|1_A0_B\rangle)]\\ \xrightarrow[]{H_1}&1/2[\alpha(|0\rangle+|1\rangle)(|0_A0_B\rangle+|1_A1_B\rangle)+\beta(|0\rangle-|1\rangle)(|0_A1_B\rangle+|1_A0_B\rangle)]\\ =&|00_A\rangle[1/2(\alpha|0_B\rangle+\beta|1_B\rangle)]\\ +&|01_A\rangle[1/2(\alpha|1_B\rangle+\beta|0_B\rangle)]\\ +&|10_A\rangle[1/2(\alpha|0_B\rangle-\beta|1_B\rangle)]\\ +&|11_A\rangle[1/2(\alpha|1_B\rangle-\beta|0_B\rangle)]\\\end{aligned}\)

3. \(A\) measures the qubits in her procession

4. \(A\) sends the information of measurement outcome via a classical channel

5. Depending on \(A\)'s message, \(B\) applies operations to his qubit and recover \(|\psi\rangle\)

   \(A\)'s outcome	\(B\)'s state	\(B\)'s operation
   |\(00_A\rangle\)	\(\alpha\)|\(0\rangle+\beta\)|\(1\rangle\)
   |\(01_A\rangle\)	\(\alpha\)|\(1\rangle+\beta\)|\(0\rangle\)	\(X\)
   |\(10_A\rangle\)	\(\alpha\)|\(0\rangle-\beta\)|\(1\rangle\)	\(Z\)
   |\(11_A\rangle\)	\(\alpha\)|\(1\rangle-\beta\)|\(0\rangle\)	\(X,Z\)

Discussion

• Does it allow fast-than-light communication?
  • No, \(A\) needs to send the outcome information via classical channel (limited by the spped of light).
• Does the protocol violate the no-cloning theorem?
  • No, at the end of the protocol, only the \(B\)'s qubit is in the state \(|\psi\rangle\)
    No information of \(|\psi\rangle\) is left on \(A\)'s qubit (only the four basis states.)
• Importance of quantum teleportation:
  1. fundamental property of quantum mechanics
  2. used e.g. in QEC
  3. measuremnt-based quantum computing
  4. distribution of quantum information in quantum networks