Quantum Key Distribution

QKD Security Framework

QKD provides information-theoretic security guaranteed by quantum mechanics. The security relies on:

1. No-cloning theorem: eavesdropper cannot copy quantum states
2. Measurement disturbance: measuring in the wrong basis introduces errors

The final secret key rate for BB84:

where is the error rate, is the binary entropy function, and is the error correction efficiency.

BB84 Protocol (Detailed)

Encoding:

Steps:

1. Alice sends qubits with random bit and basis choices
2. Bob measures each qubit in a random basis
3. Public basis reconciliation: keep only matching-basis rounds
4. Sifted key: approximately bits remain
5. Parameter estimation: publicly compare subset to estimate error rate
6. Error correction: reconcile errors (e.g., Cascade protocol)
7. Privacy amplification: compress to final key

E91 Protocol

The E91 protocol uses entanglement and Bell's inequality:

1. Source produces Bell pairs
2. Alice and Bob measure in randomly chosen bases
3. Key generation: same-basis measurements
4. Security check: different-basis measurements test Bell's inequality

The CHSH parameter:

If , Bell inequality is violated, confirming security.

Device-Independent QKD

DI-QKD provides security even when devices are untrusted:

1. Security relies only on the observed Bell violation
2. No assumptions about the internal workings of devices
3. The Bell violation certifies the presence of entanglement

DI-QKD requires high-efficiency detectors (>82% for CHSW), low-loss channels, and high-rate sources.

Continuous-Variable QKD

CV-QKD uses quadratures of light instead of photon polarization:

• GG02 protocol: uses coherent states and homodyne detection
• Advantages: compatible with standard telecom components
• Disadvantages: limited range, lower key rates at long distances

The secret key rate:

where is the reconciliation efficiency.

QKD Networks

Practical QKD networks require:

1. Trusted nodes: intermediate stations where keys are relayed
2. Quantum repeaters: enable long-distance entanglement (not yet practical)
3. Key management: key storage, distribution, and refresh

The Tokyo QKD Network and Beijing-Shanghai backbone demonstrate metropolitan and inter-city QKD.

Python: QKD Security Analysis

import numpy as np

def bb84_key_rate(n, error_rate, efficiency=0.95):
    def h(x):
        if x <= 0 or x >= 1: return 0
        return -x * np.log2(x) - (1-x) * np.log2(1-x)
    e = error_rate
    rate = 1 - h(e) - efficiency * h(e)
    return max(0, rate) * n

n = 1000000
for e in [0.01, 0.05, 0.11, 0.15]:
    rate = bb84_key_rate(n, e)
    print(f"QBER={e:.2f}: {rate:.0f} bits ({100*rate/n:.1f}%)")

correlations = [1, -1, -1, 1]
S = abs(correlations[0] - correlations[1] + correlations[2] + correlations[3])
print(f"CHSH: S = {S:.3f} (max = {2*np.sqrt(2):.3f})")

QKD Protocol Comparison