We generalize past work on quantum sensor networks to show that, for $d$ input parameters, entanglement can yield a factor $\mathcal O(d)$ improvement in mean squared error when estimating an analytic function of these parameters. We show that the protocol is optimal for qubit sensors, and conjecture an optimal protocol for photons passing through interferometers. Our protocol is also applicable to continuous variable measurements, such as one quadrature of a field operator. We outline a few potential applications, including calibration of laser operations in trapped ion quantum computing.
Heisenberg-limited eavesdropping on the continuous-variable quantum cryptographic protocol with no basis switching is impossible
基于相干态和外差检测的高斯量子密钥分发协议[Phys. Rev. Lett. 93，170504（2004）]具有在接收器侧不需要主动随机基切换的优点。它的安全性，但是，不是很令人满意的理解今天，因为已从海森堡关系推导出的秘密密钥速率的界限是没有达到任何已知的计划。在这里，我们解决了这个协议的最佳高斯个人攻击的问题，并推导出严格的上限上的信息窃听。最后给出了达到该界的最优方案，并对协议的安全性进行了分析。
Super-Heisenberg protocol for dark matter and high-frequency gravitational wave search
Tokyo Metropolitan University（东京都立大学）；KEK Theory Center（高能加速器研究机构理论中心）；The Graduate University for Advanced Studies (SOKENDAI)（综合研究大学院大学）；Yukawa Institute for Theoretical Physics, Kyoto University（京都大学汤川理论物理研究所）
Butterfly Echo Protocol for Axis-Agnostic Heisenberg-Limited Metrology
United States Naval Academy（美国海军学院）；Harvard University（哈佛大学）；Massachusetts Institute of Technology（麻省理工学院）；The College of William & Mary（威廉玛丽学院）；NIST/University of Maryland（美国国家标准与技术研究院/马里兰大学）
Heisenberg-limited metrology in the presence of non-Markovian noise with finite control rates
Center for Quantum Information and Control, University of New Mexico（新墨西哥大学量子信息与控制中心）；Department of Physics and Astronomy, University of New Mexico（新墨西哥大学物理与天文学系）
Learning to Restore Heisenberg Limit in Noisy Quantum Sensing via Quantum Digital Twin
Shanghai Jiao Tong University（上海交通大学）；Hefei National Laboratory（合肥国家实验室）；Shanghai Research Center for Quantum Sciences（上海量子科学研究中心）；Lenovo Research（联想研究院）
A generalized echo squeezing protocol with near-Heisenberg limit sensitivity and strong robustness against excess noise and variation in squeezing parameter
Heisenberg-limited Bayesian phase estimation with low-depth digital quantum circuits
California Institute of Technology（加州理工学院）；School of Physics and Astronomy, Tel Aviv University（特拉维夫大学物理与天文学学院）；Racah Institute of Physics, The Hebrew University of Jerusalem（耶路撒冷希伯来大学拉卡物理研究所）
Encoded Quantum Signal Processing for Heisenberg-Limited Metrology
Physical Detection Systems and Deployment Division, Pacific Northwest National Laboratory（太平洋西北国家实验室物理探测系统与部署部）；Department of Computer Science, Colorado State University（科罗拉多州立大学计算机科学系）；Department of Physics, University of Toronto（多伦多大学物理系）；Advanced Computing, Mathematics, and Data Division, Pacific Northwest National Laboratory（太平洋西北国家实验室先进计算、数学与数据部）
Continuous Variable Hamiltonian Learning at Heisenberg Limit via Displacement-Random Unitary Transformation
School of Stomatology, Peking University（北京大学口腔医学院）；Department of Chemistry and Biochemistry, University of California, Los Angeles（加州大学洛杉矶分校化学与生物化学系）；Department of Physics, Tsinghua University（清华大学物理系）；Institute of Advanced Study, Tsinghua University（清华大学高等研究院）
