One-Sided Device-Independent Quantum Key Distribution over Noisy Metropolitan Links: Noise Thresholds and Purification-Assisted Recovery

One-sided device-independent quantum key distribution (1SDI-QKD) offers a practical middle ground between fully device-independent protocols and standard QKD, achieving security with detection efficiencies as low as 50.1% on the untrusted side. However, prior analyses assumed idealized channels. This work extends the 1SDI-QKD framework to amplitude damping, dephasing, and depolarizing noise. Results reveal a clear noise hierarchy: dephasing tolerates the most noise before security is lost (p crit about 0.50 versus 0.449 and 0.220 respectively), while depolarizing noise imposes the most stringent detection efficiency requirements, exceeding 97% at 20% noise where amplitude damping requires 76%. Security is lost while substantial entanglement remains (C about 0.67 -- 0.74), demonstrating that steering violation, not merely entanglement, determines 1SDI-QKD security. Integrating the BBPSSW purification protocol shows that it functions primarily as a range-extension mechanism: five rounds extend dephasing-limited operation from 10 km to 36 km at 90% detection efficiency. Accounting for both the 2 n pair cost and the cumulative success probability, the effective rate peaks at 10-3 bits per initial pair for dephasing but only 10-6 for amplitude damping and depolarizing noise, identifying phase noise limited links as the regime where purification is worthwhile.