Table of Contents
- Introduction
- Historical Context
- Core Concepts Behind E91
- E91 vs BB84
- Entanglement in E91
- Bell’s Theorem and Non-Locality
- Step-by-Step E91 Protocol
- The Role of Bell Inequalities
- The CHSH Inequality
- Security Mechanism in E91
- Quantum States Used
- Measurement Settings
- Detecting Eavesdropping
- Mathematical Framework
- Key Extraction Process
- Classical Communication and Post-Processing
- Advantages of E91
- Practical Implementations
- Experimental Demonstrations
- Limitations and Technical Challenges
- Device Independence in E91
- Application in Quantum Networks
- E91 and Future Satellite QKD
- Comparison Summary with Other Protocols
- 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
| Feature | BB84 | E91 |
|---|---|---|
| Resource | Single qubits | Entangled pairs |
| Security basis | Basis mismatch | Bell inequality violation |
| Implementation | Simpler | More complex |
| Device independence | No | Yes (with assumptions) |
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
- A central source emits entangled photon pairs to Alice and Bob.
- Each randomly selects a measurement setting (3 each).
- They perform measurements on their qubits.
- They compare settings publicly.
- Use measurement results with matching settings to form the secret key.
- 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
| Protocol | Basis | Uses Entanglement | Device Independent |
|---|---|---|---|
| BB84 | Basis choice | No | No |
| E91 | Bell test | Yes | Yes (potentially) |
| B92 | Simplified states | No | No |
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.
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