Understanding Quantum Teleportation
Quantum teleportation is one of the most striking applications of quantum entanglement. It allows you to transfer an unknown quantum state from one location (Alice) to another (Bob) using shared entanglement and classical communication. Crucially, the protocol demonstrates three key quantum principles: entanglement as a resource, measurement-induced collapse, and the no-cloning theorem.
The Protocol
Step 1: Shared Entanglement
Alice and Bob share a Bell pair |Φ⁺⟩ = (|00⟩ + |11⟩)/√2. This is an entangled state of two qubits prepared beforehand. Alice has one qubit of the pair; Bob has the other. This entangled resource is what makes teleportation possible—without it, no classical protocol could transfer an unknown quantum state.
Step 2: Alice's Unknown State
Alice has a qubit in state |ψ⟩ = cos(θ/2)|0⟩ + eiφsin(θ/2)|1⟩ that she wants to send to Bob. She doesn't know what θ and φ are (the state is unknown to her). The three-qubit system is now: |ψ⟩ ⊗ |Φ⁺⟩. Alice's first qubit is the input; her second qubit is from the Bell pair; Bob's qubit is the other half of the Bell pair.
Step 3: Bell Measurement
Alice performs a joint measurement of her two qubits in the Bell basis. This projects the two qubits onto one of four Bell states: |Φ⁺⟩, |Φ⁻⟩, |Ψ⁺⟩, |Ψ⁻⟩. Each outcome occurs with probability 1/4. The measurement collapses Alice's qubits and simultaneously transforms Bob's qubit into a rotated version of |ψ⟩. Crucially, Alice's original state is destroyed—she cannot keep a copy (no-cloning theorem).
Step 4: Classical Communication
Alice sends Bob two classical bits telling him which of the four Bell states she measured. This classical channel is essential: without it, Bob's qubit is in a maximally mixed state (complete ignorance). Quantum teleportation does not enable faster-than-light communication—Bob learns nothing about |ψ⟩ until the classical bits arrive.
Step 5: Bob's Correction
Based on Alice's two bits, Bob applies one of four Pauli corrections: I (do nothing), X (bit flip), Z (phase flip), or XZ (both). After this correction, Bob's qubit is exactly in state |ψ⟩. The teleportation is complete. Bob now has the state Alice started with, but Alice no longer has it—the state was moved, not copied.
Key Corrections
The Four Measurement Outcomes
Depending on Alice's Bell measurement outcome, Bob's state is related to |ψ⟩ by a Pauli operation. The mapping is:
- |Φ⁺⟩ (bits 00): Bob has |ψ⟩ → Apply I (identity, do nothing)
- |Φ⁻⟩ (bits 01): Bob has Z|ψ⟩ → Apply Z to recover |ψ⟩
- |Ψ⁺⟩ (bits 10): Bob has X|ψ⟩ → Apply X to recover |ψ⟩
- |Ψ⁻⟩ (bits 11): Bob has XZ|ψ⟩ → Apply ZX to recover |ψ⟩
Each outcome is equally likely (probability 1/4). The entanglement ensures that Alice's measurement deterministically sets Bob's correction—this is the "spooky" correlation.
Why This Matters
No-cloning enforced: Alice's original state is destroyed during the Bell measurement. You cannot make a perfect copy of an unknown quantum state—teleportation respects this fundamental limit by moving, not duplicating.
Entanglement as a resource: The Bell pair is consumed during teleportation. Entanglement is a finite resource that enables tasks impossible with classical correlations alone.
Classical channel required: Bob's qubit is maximally mixed before he receives Alice's classical bits. This prevents faster-than-light communication—no information is transmitted superluminally.
Perfect fidelity: Quantum teleportation has fidelity F = 1 for pure states. This is better than any classical protocol, which cannot transmit an unknown continuous state with finite classical bits.
Applications: Teleportation is used in quantum repeaters (to extend quantum communication over long distances), quantum computing (to move quantum states between processors), and quantum cryptography (to distribute quantum keys).
Things to Try
Teleport |0⟩: Set the preset to |0⟩ and step through the protocol. Notice that when Alice measures |Φ⁺⟩ (00 bits), Bob already has |0⟩—no correction needed. For other outcomes, Bob applies X, Z, or both to recover |0⟩.
Teleport |+⟩: Prepare |+⟩ = (|0⟩+|1⟩)/√2 and teleport it. Watch the Bloch spheres: Alice's state disappears after measurement, and Bob's state (after correction) matches the original. The superposition is preserved.
Random state teleportation: Choose arbitrary θ and φ. Step through the protocol and verify that fidelity F = 1 regardless of the input state. This demonstrates that teleportation works for any unknown state.
No-cloning in action: After the Bell measurement, check that Alice's first qubit is no longer in state |ψ⟩. The original state is destroyed—quantum mechanics forbids copying.
Classical bits matter: Before Bob applies the correction, his state is useless (maximally mixed). Only after receiving Alice's two classical bits can he recover |ψ⟩. This is why teleportation doesn't violate causality.
Measurement randomness: Reset and repeat the protocol multiple times with the same input state. Alice's measurement outcome changes (it's random), but Bob always ends up with the correct state after applying the correction. The randomness averages out over the four outcomes.
Historical Context
Quantum teleportation was proposed by Bennett, Brassard, Crépeau, Jozsa, Peres, and Wootters in 1993. It was first experimentally demonstrated with photons in 1997. The protocol is now a cornerstone of quantum information theory and has been implemented with various physical systems: photons, ions, superconducting qubits, and atoms. In 2017, China teleported a photon from Earth to a satellite in orbit—a distance of over 1400 km.
Despite the name, nothing physical is teleported. The quantum state—the information encoded in θ and φ—is transferred using entanglement and classical bits. The original qubit remains at Alice's location, but its state is destroyed. This is fundamentally different from science fiction teleportation, but it's still remarkable: you can send an unknown quantum state without ever knowing what it is.