Temporal Side-Channel Vulnerability of Decoy-State Quantum Key Distribution under Quantum Non-Demolition Eavesdropping

Quantum key distribution promises information-theoretic security, but practical implementations are vulnerable to side-channel attacks that exploit unintended non-operational physical degrees of freedom. A particularly subtle threat arises from the temporal distinguishability of signal and decoy pulses in decoy-state BB84 protocol. If an eavesdropper can discriminate quantum state by its timing, she can tag the intensity class and break the central assumption of the decoy-state method. In this work, we analyze a temporal side-channel attack in which the eavesdropper employs a realistically imperfect quantum non-demolition measurement module to extract timing information without absorbing the photon. We show that the maximum-likelihood discrimination probability of signal and decoy pulses reaches 0.68 after 60 km of standard single-mode fiber, well above the random-guessing baseline of 0.5. With the quantum non-demolition module placed at 70 km, the discrimination probability increases to 0.70--0.73, depending on the interaction time, while at short distances (< 5 km) it remains nearly independent of the quantum non-demolition parameters. The secure key rate, evaluated within a modified GLLP framework, drops to zero beyond 36 km in both cases---with the maximum-likelihood discrimination alone and with the imperfect-quantum non-demolition module---differing only in overall efficiency. To obtain these results, we develop a comprehensive model that combines realistic fiber propagation with a non-ideal quantum non-demolition interaction and statistical decision theory. The proposed methodology provides a practical toolbox for quantifying temporal side-channel vulnerabilities and can serve as a foundation for future security analyses of quantum key distribution systems.