Quantum Teleportation

Table of Contents

1. Introduction
2. What Is Quantum Teleportation?
3. Misconceptions About Teleportation
4. Why Teleportation Matters
5. Basic Ingredients of Quantum Teleportation
6. The Quantum Circuit
7. Step-by-Step Protocol
8. The Role of Entanglement
9. Mathematical Derivation
10. Measurement and Classical Communication
11. Reconstruction of the Original State
12. Teleportation vs Cloning
13. No-Cloning Theorem and Teleportation
14. Resources Required
15. Bell Basis Measurement
16. Teleportation of Mixed States
17. Fidelity of Teleportation
18. Experimental Realizations
19. Teleportation with Photons
20. Teleportation in Ion Traps and Superconducting Qubits
21. Long-Distance Quantum Communication
22. Quantum Repeaters
23. Role in Quantum Networks
24. Limitations and Challenges
25. Conclusion

1. Introduction

Quantum teleportation is a process by which the state of a quantum particle is transferred from one location to another, using entanglement and classical communication. The particle itself is not physically moved, but its quantum state is recreated elsewhere.

2. What Is Quantum Teleportation?

Teleportation transfers an unknown quantum state \( |\psi\rangle \) from a sender (Alice) to a receiver (Bob), using:

• An entangled pair shared between Alice and Bob
• A Bell state measurement by Alice
• Two classical bits of communication

3. Misconceptions About Teleportation

Quantum teleportation:

• Does not transport matter
• Does not violate special relativity
• Requires classical communication, hence not instantaneous

4. Why Teleportation Matters

Quantum teleportation is essential for:

• Quantum networks
• Distributed quantum computing
• Quantum cryptography
• Quantum repeaters for long-distance communication

5. Basic Ingredients of Quantum Teleportation

• A qubit \( |\psi\rangle = \alpha|0\rangle + \beta|1\rangle \) to teleport
• An entangled Bell pair \( |\Phi^+\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle) \)
• Alice and Bob at different locations

6. The Quantum Circuit

Teleportation involves the following gates:

• CNOT
• Hadamard
• Measurement
• Conditional X and Z gates

7. Step-by-Step Protocol

1. Alice prepares \( |\psi\rangle \) and shares a Bell state with Bob.
2. She performs a Bell measurement on her qubits.
3. She sends 2 classical bits to Bob.
4. Bob applies X and Z corrections based on Alice’s message.
5. Bob recovers \( |\psi\rangle \).

8. The Role of Entanglement

Entanglement is the quantum resource that enables teleportation. Without it, the protocol is impossible, regardless of how much classical data is shared.

9. Mathematical Derivation

Let \( |\psi\rangle = \alpha|0\rangle + \beta|1\rangle \), and \( |\Phi^+\rangle_{AB} \) be the entangled pair.

The combined state:

\[
|\psi\rangle_1 \otimes |\Phi^+\rangle_{23}
= \frac{1}{\sqrt{2}} (\alpha|0\rangle_1 + \beta|1\rangle_1)(|00\rangle_{23} + |11\rangle_{23})
\]

This expands and is rewritten in the Bell basis for Alice’s measurement, collapsing Bob’s qubit into a state that can be corrected into \( |\psi\rangle \).

10. Measurement and Classical Communication

Alice’s Bell measurement projects her two qubits into one of four Bell states. She then sends two classical bits corresponding to this outcome to Bob.

11. Reconstruction of the Original State

Based on Alice’s result:

• 00 → do nothing
• 01 → apply \( X \)
• 10 → apply \( Z \)
• 11 → apply \( XZ \)

Bob’s qubit becomes \( |\psi\rangle \).

12. Teleportation vs Cloning

Quantum teleportation destroys the original — it’s not copying. This is in full compliance with the no-cloning theorem.

13. No-Cloning Theorem and Teleportation

Teleportation respects:

\[
\text{No quantum information is copied.}
\]

The process involves destruction of the original state through measurement.

14. Resources Required

• One maximally entangled pair
• Two classical bits
• Measurement and conditional gate application

15. Bell Basis Measurement

Bell basis states:

\[
|\Phi^\pm\rangle = \frac{1}{\sqrt{2}}(|00\rangle \pm |11\rangle), \quad
|\Psi^\pm\rangle = \frac{1}{\sqrt{2}}(|01\rangle \pm |10\rangle)
\]

Bell measurement projects a state into one of these.

16. Teleportation of Mixed States

Teleportation also works (with fidelity loss) for mixed states using density matrices and Kraus operators.

17. Fidelity of Teleportation

Fidelity \( F \) quantifies how close the received state is to the original:

\[
F = \langle \psi|\rho_{\text{out}}|\psi\rangle
\]

Perfect teleportation: \( F = 1 \)

18. Experimental Realizations

• Photons using beam splitters and detectors
• Ions in traps using laser pulses
• Superconducting qubits with microwave links
• NV centers in diamond

19. Teleportation with Photons

Photonic teleportation uses polarization qubits, entangled photon sources, and Bell state analyzers.

20. Teleportation in Ion Traps and Superconducting Qubits

Gate-based systems perform teleportation using controlled gate sequences and microwave pulse timing.

21. Long-Distance Quantum Communication

Teleportation enables:

• Quantum key distribution
• Entanglement distribution
• Quantum internet infrastructure

22. Quantum Repeaters

Combat decoherence in long-distance links by:

• Dividing into segments
• Teleporting entangled pairs through intermediate stations

23. Role in Quantum Networks

Teleportation enables quantum routers and distributed quantum computing, where qubits are physically separated but logically connected.

24. Limitations and Challenges

• Entanglement fidelity limits teleportation fidelity
• Loss in optical channels
• Detection inefficiencies
• Bell measurement success rates below 100%

25. Conclusion

Quantum teleportation is a cornerstone of quantum information science. It demonstrates the power of entanglement, classical communication, and quantum measurements working together to transmit quantum states. Far from science fiction, teleportation is a real, experimentally verified process that underpins future quantum networks and secure communication systems.