Analysis of the BB84 Quantum Key Distribution (QKD) Algorithm

Overview

In this notebook, we implement and analyze the BB84 Quantum Key Distribution (QKD) protocol under a variety of conditions to study how eavesdropping and environmental noise affect the ability of Alice and Bob to securely generate a shared key. We simulate multiple cases, visualize quantum circuits, and apply post-processing techniques including error correction and privacy amplification.

Simulation Cases

We model four scenarios to test how QBER and key generation behave under different conditions:

1. No Eavesdropping, Low Noise
   Ideal conditions. Expected QBER ≈ 0.

2. Eavesdropping, Low Noise
   Eve performs measurements on the quantum channel. Expected QBER ≈ 0.25.

3. No Eavesdropping, High Noise
   Channel noise flips qubits randomly. Expected QBER ≈ noise level.

Each case includes:

• Circuit construction and visualization
• Statistical analysis across multiple trials
• Comparison to theoretical expectations

Sifted Key Generation

After measurement, Alice and Bob compare their basis choices and retain only bits where their bases match. This forms the sifted key, which is used for further processing. We track:

• Sifted key length
• Number of discarded runs
• QBER per trial

Key Rate and QBER Analysis

We calculate the secure key rate using the binary entropy function:

$$H(e) = -e \log_2 e - (1 - e) \log_2 (1 - e)$$ $$R = \max(0,\ 1 - 2H(e))$$

This quantifies how much usable key material remains after error correction and privacy amplification. We plot QBER and key rate vs. noise level with and without Eve.

LDPC Error Correction

We apply Low-Density Parity-Check (LDPC) codes to correct mismatches in the sifted key. Using pyldpc, we:

• Encode Alice’s key
• Simulate transmission with QBER-induced bit flips
• Decode Bob’s received word
• Recover the corrected key

Privacy Amplification

To ensure information-theoretic security, we hash the corrected key using SHA-256. This produces a final secure key that is resistant to both classical and quantum attacks.

Final Outcome

The notebook demonstrates a complete BB84 pipeline:

• From quantum circuit simulation
• To sifted key extraction
• To secure key generation

It validates BB84’s robustness against noise and eavesdropping, and shows how post-processing techniques enable secure communication even under imperfect conditions.