[Quantum Information] Quantum teleportation

  Quantum teleportation

Notes from RWTH Aachen University course 
“Quantum Information” Summer semester 2020
professor: Müller, Markus

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

Readings

Bennett, Charles & Brassard, Gilles & Crépeau, Claude & Jozsa, Richard & Peres, Asher & Wootters, William. (1993). Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels. Physical review letters. 70. 1895-1899. 10.1103/PhysRevLett.70.1895.