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    Bayesian Phase Stabilization at the Shot-Noise Limit for Scalable Quantum Networks

    Guang-Cheng Liu1,3,4,*, Chao-Hui Xue1,2,3,*, Fa-Xi Chen2,3,*, Ming-Yang Zheng2,3, Yi Yang1,2,3, Li-Bo Li2, Bin Wang2, Bo-Wen Yang1,3,4, Hai-Feng Jiang1,3,4 et al.

    Yong Wan1,3,4, Ye Wang1,3,4, Jiu-Peng Chen2,3, Qiang Zhang1,3,2,4, and Jian-Wei Pan1,3,4

    • *These authors contributed equally to this work.

    Phys. Rev. Lett. 137, 160802 – Published 14 October, 2026

    DOI: https://doi.org/10.1103/cxs1-3pzf

    Abstract

    High-precision optical phase stabilization in quantum networks is fundamentally constrained by the strict photon-flux and duty-cycle limits required to avoid disturbing fragile quantum states. This challenge becomes especially critical when coordinating multiple independent light sources for multistep quantum protocols. Here, we develop an integrated phase-stabilization framework that incorporates a Bayesian phase estimator to optimally extract information from sparse single-photon detection events. This approach outperforms conventional maximum-likelihood estimation and achieves the shot-noise limit under minimal photon flux. The framework enables real-time correction of combined phase noise from both nodal lasers and transmission fibers, facilitating a two-step excitation protocol for heralded entanglement generation between separate trapped-ion nodes via single-photon interference. Operating with a detected photon rate of ∼1  MHz and a duty cycle ≤6.5%, the system maintains interferometric visibility >97% over fiber links of 10 km and 100 km. This phase control yields long-lived ion-ion entanglement with parity contrast exceeding 85%, enabling device-independent quantum key distribution with finite-size security analysis over 10 km and a positive asymptotic key-rate over 100 km. Moreover, the resulting memory-memory entanglement at 10 km survives beyond the average time required to establish it—a fundamental requirement for quantum repeaters. This Letter establishes a robust and scalable foundation for practical long-distance quantum networks.

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