E91 Protocol

Table of Contents

1. Introduction
2. Historical Context
3. Core Concepts Behind E91
4. E91 vs BB84
5. Entanglement in E91
6. Bell’s Theorem and Non-Locality
7. Step-by-Step E91 Protocol
8. The Role of Bell Inequalities
9. The CHSH Inequality
10. Security Mechanism in E91
11. Quantum States Used
12. Measurement Settings
13. Detecting Eavesdropping
14. Mathematical Framework
15. Key Extraction Process
16. Classical Communication and Post-Processing
17. Advantages of E91
18. Practical Implementations
19. Experimental Demonstrations
20. Limitations and Technical Challenges
21. Device Independence in E91
22. Application in Quantum Networks
23. E91 and Future Satellite QKD
24. Comparison Summary with Other Protocols
25. Conclusion

1. Introduction

The E91 Protocol, proposed by Artur Ekert in 1991, is a quantum key distribution (QKD) scheme based on the principles of quantum entanglement and Bell’s theorem. Unlike BB84, E91 uses entangled particles to ensure security and detect eavesdropping.

2. Historical Context

The E91 protocol introduced the idea that quantum correlations verified by Bell inequality violations could be used to distribute cryptographic keys securely. It connected quantum information theory with the foundations of quantum mechanics.

3. Core Concepts Behind E91

• Uses maximally entangled pairs (e.g., Bell states)
• Security stems from violation of Bell inequalities
• Eavesdropping disrupts the quantum correlations and changes measurement statistics

4. E91 vs BB84

5. Entanglement in E91

Pairs of qubits are generated in a Bell state:

\[
|\Phi^+\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)
\]

One qubit is sent to Alice and the other to Bob.

6. Bell’s Theorem and Non-Locality

Bell’s theorem shows that no local hidden variable theory can reproduce the predictions of quantum mechanics. In E91, violation of Bell inequalities implies genuine quantum correlations.

7. Step-by-Step E91 Protocol

1. A central source emits entangled photon pairs to Alice and Bob.
2. Each randomly selects a measurement setting (3 each).
3. They perform measurements on their qubits.
4. They compare settings publicly.
5. Use measurement results with matching settings to form the secret key.
6. Other settings are used to test the CHSH Bell inequality.

8. The Role of Bell Inequalities

Violating Bell inequalities ensures:

• The quantum correlations are non-classical
• No eavesdropper can simulate them without detection

9. The CHSH Inequality

The Clauser-Horne-Shimony-Holt (CHSH) version is:

\[
|E(a, b) + E(a, b’) + E(a’, b) – E(a’, b’)| \leq 2
\]

Quantum mechanics allows values up to \( 2\sqrt{2} \). Violation signals quantum entanglement.

10. Security Mechanism in E91

Any eavesdropper trying to intercept or replicate the entangled states will:

• Alter the statistics
• Reduce Bell inequality violation
• Be detected by Alice and Bob

11. Quantum States Used

E91 uses Bell states, like:

\[
|\Psi^-\rangle = \frac{1}{\sqrt{2}}(|01\rangle – |10\rangle)
\]

which have perfect anti-correlations in measurement outcomes.

12. Measurement Settings

Alice uses: \( A_1, A_2, A_3 \)
Bob uses: \( B_1, B_2, B_3 \)

Certain combinations are used for:

• Bell inequality tests
• Key extraction

13. Detecting Eavesdropping

If CHSH inequality is not violated, it means:

• Eavesdropper tampered with the entangled states
• The communication is not secure

14. Mathematical Framework

Expected correlations are computed as:

\[
E(a, b) = P_{++}(a, b) + P_{–}(a, b) – P_{+-}(a, b) – P_{-+}(a, b)
\]

Where \( P_{ij}(a, b) \) is the joint probability of outcomes \( i \) and \( j \).

15. Key Extraction Process

• Only a subset of results is used to form the key.
• Results corresponding to aligned measurement bases form the sifted key.

16. Classical Communication and Post-Processing

Alice and Bob:

• Share basis choices
• Estimate error rates
• Perform privacy amplification and error correction

17. Advantages of E91

• More secure against side-channel attacks
• Can be made device-independent
• Based on deeper quantum principles

18. Practical Implementations

• Photonic entanglement over fiber optics
• Free-space optical experiments
• Real-time CHSH inequality testing

19. Experimental Demonstrations

E91-like QKD systems have been tested over:

• Urban fiber networks
• Satellite links (e.g., Micius satellite by China)
• Long-distance entanglement distribution (>1200 km)

20. Limitations and Technical Challenges

• Entangled sources are complex
• Synchronization required between distant parties
• Photon loss and detector inefficiencies

21. Device Independence in E91

By violating Bell inequalities, E91 allows for device-independent security, reducing trust requirements in hardware.

22. Application in Quantum Networks

Entanglement-based protocols like E91 are foundational for:

• Quantum internet
• Entanglement swapping
• Quantum repeaters

23. E91 and Future Satellite QKD

E91 is ideal for space-based QKD due to:

• Long-range entanglement
• Robustness to noise
• Fundamental tests of quantum mechanics

24. Comparison Summary with Other Protocols

25. Conclusion

The E91 protocol represents a landmark in quantum communication, showing how the non-locality of entanglement can be harnessed for unbreakable cryptographic key distribution. Though more complex than BB84, its potential for device independence, long-distance QKD, and quantum internet applications make it a central pillar of future secure communications.