Quantum Teleportation

A protocol to transfer quantum states using entanglement and classical communication, without physically sending the quantum system.

Quantum teleportation transmits an unknown quantum state from one location to another using a shared entangled pair and classical communication. No physical quantum system travels; only classical information is sent.

The Setup

Three parties:

Alice: Has unknown state $∣ ψ ⟩$ to teleport
Bob: Will receive the state
Shared resource: Alice and Bob share a Bell pair

The Protocol

Initial State

$∣ ψ ⟩_{A} \otimes ∣ Φ^{+} ⟩_{A^{'} B} = ∣ ψ ⟩_{A} \otimes \frac{1}{2} (∣00 ⟩ + ∣11 ⟩)_{A^{'} B}$

Alice has qubit $A$ (the unknown state) and qubit $A^{'}$ (her half of the Bell pair). Bob has qubit $B$ (his half of the Bell pair).

Step 1: Bell Measurement

Alice performs a Bell state measurement on qubits $A$ and $A^{'}$ , getting one of four outcomes: 00, 01, 10, or 11.

Step 2: Classical Communication

Alice sends her 2-bit measurement result to Bob (classical channel).

Step 3: Correction

Based on Alice’s message, Bob applies a correction:

Alice’s Result	Bob’s Correction
00	$I$ (nothing)
01	$X$
10	$Z$
11	$XZ$

Result

Bob now has $∣ ψ ⟩$ exactly. Alice’s original is destroyed.

Circuit Diagram

|ψ⟩ ────●────H────M──────────╓
        │                    ║ (classical)
|Φ+⟩ ───⊕─────────M────────╫─╫──
                           ║ ║
|B⟩ ───────────────────X───╫─Z── |ψ⟩
                       ▲   ║
                       └───╨───

Key Properties

No Faster-Than-Light Communication

Teleportation requires classical communication (limited to light speed). Without knowing Alice’s measurement result, Bob’s qubit looks completely random.

Destroys Original State

The no-cloning theorem is satisfied: $∣ ψ ⟩$ is destroyed at Alice’s location when created at Bob’s.

Requires Pre-Shared Entanglement

The Bell pair must be distributed beforehand. Teleportation “spends” the entanglement.

Applications

Application	Use
Quantum networks	Transfer quantum states between nodes
Quantum computing	Move states between processors
Quantum repeaters	Enable long-distance quantum communication
Gate teleportation	Implement gates using teleportation

Gate Teleportation

A powerful generalization: instead of teleporting $∣ ψ ⟩$ , teleport $U ∣ ψ ⟩$ for some gate $U$ . This is key to fault-tolerant quantum computing.

Experimental Milestones

1997: First demonstration (photons, ~1 meter)
2012: 143 km (Canary Islands)
2017: 1,400 km (satellite, Micius)
Ongoing: Integrated into quantum networks

See also: Entanglement, Bell State, No-Cloning Theorem

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