# 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.)
- <ins>Importance</ins> 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