Researchers at the Hefei National Laboratory and University of Science and Technology of China have developed a phase-stabilization framework that achieves the shot-noise limit, a fundamental barrier in quantum communication. The system uses a Bayesian phase estimator to extract information from sparse single-photon detection events, outperforming conventional methods while operating with a detected photon rate of approximately 1 million per second and a duty cycle of 6.5 percent or less. This phase control maintains interferometric visibility exceeding 97% over fiber links of 10 and 100 kilometers, enabling long-lived ion-ion entanglement and device-independent quantum key distribution.

Bayesian Estimation Surpasses Limits of Photon Flux

This achievement circumvents a longstanding trade-off between precision and delay inherent in conventional maximum-likelihood estimation (MLE), which struggles when tracking rapidly evolving phases with limited photon flux. The new framework minimizes the photon budget and duty cycle of reference probes, a critical feature for preserving fragile quantum states essential for scalable networks. Conventional MLE faces limitations because extended integration, while improving accuracy, accumulates unbounded phase diffusion from environmental disturbances; rapid sampling, conversely, is limited by the inherent noise of individual photons.

This correction facilitated a two-step excitation protocol for generating heralded entanglement between separate trapped-ion nodes through single-photon interference. 5 percent. This level of stability is achieved through a three-dimensional representation of residual phase variance, demonstrating that the Bayesian estimation approach significantly outperforms conventional MLE, particularly at lower photon fluxes. The framework’s resilience to correlated phase noise further highlights its potential for practical, real-world deployment.

The Bayesian approach analyzes phase tracking for a time-varying phase within a Mach-Zehnder interferometer probed by weak coherent states, calculating mean photon counts at the output ports based on photon flux and interference visibility. These calculations yield a classical Fisher information, which is then used to optimize the phase estimation process. The system uses acousto-optic modulators (AOMs) in the interferometer arms, controlled by an FPGA-based servo controller, to apply corrective feedback based on time-stamped photon-detection events.

Etalons placed in each output arm further suppress nonlinear noise from the quantum frequency conversion (QFC) process. The framework’s photon efficiency extends beyond quantum networking, offering potential benefits for astronomical interferometry, lengthening observational baselines for faint sources, and precision metrology, enhancing phase tracking in low-signal conditions. Future work, they suggest, could involve integrating machine-learning techniques to dynamically adapt the phase-diffusion prior, further suppressing technical imperfections and extending robust operation to dynamic field environments.

This research received support from ZR2022LLZ006 and ZR2022LLZ011, and was led by Guang-Cheng Liu, Chao-Hui Xue and Fa-Xi Chen, who contributed equally to the work. The team’s success in reaching the shot-noise limit represents a step toward building practical and scalable quantum networks capable of secure communication and distributed quantum computation.

Two-Step Entanglement via Stabilized Single-Photon Interference

A Bayesian estimation framework now allows for the correction of combined phase noise from both nodal lasers and transmission fibers, a critical advancement for multi-step quantum protocols. The core of this improvement lies in the Bayesian phase estimator’s ability to combine prior knowledge of phase-diffusion dynamics with limited photon detections, effectively increasing the Fisher information beyond what traditional methods achieve.

Classical Fisher information, used to optimize phase estimation, is enhanced by this approach, allowing for more precise tracking of phase variations within a Mach-Zehnder interferometer. Calculations reveal that incorporating persistent prior information breaks the conventional trade-off between precision and delay, resulting in a steady-state variance that scales inversely with the square root of the photon flux, a hallmark of the shot-noise limit.

Experimental validation of the framework involved two trapped-ion nodes, each housing a single 40Ca+ ion, separated by a 3-meter free-space baseline. To simulate independent laser systems susceptible to environmental fluctuations, beams were delivered through separate 20-meter fibers, introducing uncorrelated phase noise between the nodes. The system demonstrated interferometric visibility of 97.98% ± 0.31% over a 10 kilometer fiber link and 97.49% ± 0.66% over a 100 kilometer fiber link.

These observed visibilities, while slightly below the theoretical system limit, confirm the efficacy of the Bayesian approach in real-world conditions. The framework’s performance directly enables high-fidelity memory-memory entanglement generation, a fundamental requirement for scalable quantum repeaters.

10km & 100km Fiber Links Maintain 97% Visibility

5%. This level of performance, achieved through Bayesian phase estimation, facilitates long-lived entanglement between ions with a parity contrast exceeding 85% at both distances. The resulting stability enables device-independent quantum key distribution, providing finite-size security analysis over 10 kilometers and a positive asymptotic key-rate.

Researchers characterized phase diffusion between nodes by measuring diffusion coefficients for individual fiber spans and total residual phase noise using continuous-wave probes, establishing Bayesian priors for subsequent experiments. Laboratory environmental fluctuations, primarily acoustic and thermal coupling below 200 Hz, induce frequency deviations that accumulate over time, but the system’s design mitigates these effects.

Measured phase variance scales as $σ_{T}^{2}$ ∝ $T^{1.4}$ over extended durations T, indicating correlated, non-white noise, yet the system maintains phase stability. Experimental results confirm interferometric visibility exceeding 97.7% over 10-kilometer fiber links, achieved with a 500 microsecond experimental sequence and a 32.5 microsecond probe duration at 393 nanometers. Similarly, the system maintains 97% visibility over 100-kilometer links, using an 800 microsecond sequence and a 47.5 microsecond probe duration.

These measurements were conducted under realistic laboratory noise conditions, demonstrating the robustness of the Bayesian estimation framework. The framework’s performance relies on acousto-optic modulators, dense wavelength-division multiplexing, superconducting nanowire single-photon detectors, quantum frequency conversion and periodically poled lithium niobate, integrated into a system controlled by a field-programmable gate array. The approach diverges from conventional maximum-likelihood estimation by incorporating a prior informed by the observed non-white noise characteristics.

This prior, established through continuous-wave pre-characterization, allows the estimator to assess the statistical consistency of each measurement innovation against a threshold multiplier κ. The system’s mathematical model implies a variance scaling $σ_{T}^{2}$ = kDτ over a duration T = kτ, establishing the prior for optimal tracking. This advancement, coupled with the high fidelity of the generated entanglement, positions the system as a promising platform for future quantum technologies and secure communication protocols.

Long-Lived Entanglement Enables Quantum Repeaters

5%, demonstrating shot-noise limited performance even under these minimal conditions. The experimental setup features a dual-band architecture incorporating two parallel Bayesian phase estimators, actively correcting phase noise from both lasers and fiber transmission. Time-division multiplexing synchronizes the process with the entanglement-generation sequence, interleaving weak 393 nanometer phase-probe pulses during experimental dead times. “The system maintains shot-noise limited performance with > 97% interferometric visibility over 10 km and 100 km fiber links,” the researchers report, demonstrating a quantum-optimal tracking performance confirmed by variance scaling as the inverse square root of the photon flux.

Measurements demonstrate that the residual phase variance follows theoretical predictions, confirming the system’s ability to operate at the shot-noise limit even with limited photon availability. The high-fidelity phase control enables deterministic ion-ion entanglement generation with a measured parity contrast exceeding 85% at an excitation probability of 5%, meeting the stringent requirements for scalable quantum repeaters. This work establishes a robust and scalable foundation for practical long-distance quantum networks.

Phase Control Supports Device-Independent Key Distribution

Maintaining stable quantum links, even over relatively short distances, demands precise control of optical phase, and a newly demonstrated system achieves this at the fundamental shot-noise limit. This level of performance is critical for enabling device-independent quantum key distribution, a method of secure communication that relies on the laws of physics rather than the security of the devices themselves. This approach is particularly advantageous when coordinating multiple independent light sources, as is necessary for multi-step quantum protocols.

Beyond secure communication, the system’s capabilities also address a fundamental challenge in building quantum repeaters. The resulting memory-memory entanglement at 10 kilometers persists longer than the average time needed to establish it, a significant characteristic for extending the range of quantum networks.

This persistence is a direct consequence of the precise phase stabilization, which minimizes decoherence and allows entanglement to be maintained for a sufficient duration to be used in repeater protocols. The system’s design, incorporating time-division multiplexing, synchronizes the entanglement-generation sequence with microsecond-duration probe pulses, further enhancing stability. The system’s performance demonstrates shot-noise limited operation even under photon-starved conditions, a significant advancement over previous methods.

👉 More information
🗞 Bayesian Phase Stabilization at the Shot-Noise Limit for Scalable Quantum Networks
✍️ Guang-Cheng Liu et al.
🧠 DOI: http://link.aps.org/doi/10.1103/cxs1-3pzf

More like this