Private and Robust States for Distributed Quantum Sensing

Luís Bugalho; Majid Hassani; Yasser Omar; Damian Markham

doi:10.22331/q-2025-01-15-1596

Private and Robust States for Distributed Quantum Sensing

Luís Bugalho^1,2,3,4, Majid Hassani⁴, Yasser Omar^1,2,3, and Damian Markham⁴

¹Instituto Superior Técnico, Universidade de Lisboa, Portugal
²Physics of Information and Quantum Technologies Group, Centro de Física e Engenharia de Materiais Avançados (CeFEMA), Portugal
³PQI – Portuguese Quantum Institute, Portugal
⁴Sorbonne Université, CNRS, LIP6, 4 Place Jussieu, Paris F-75005, France

Published:	2025-01-15, volume 9, page 1596
Editor:	Christos Gagatsos
Eprint:	arXiv:2407.21701v2
Doi:	https://doi.org/10.22331/q-2025-01-15-1596
Citation:	Quantum 9, 1596 (2025).

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Abstract

Distributed quantum sensing enables the estimation of multiple parameters encoded in spatially separated probes. While traditional quantum sensing is often focused on estimating a single parameter with maximum precision, distributed quantum sensing seeks to estimate some function of multiple parameters that are only locally accessible for each party involved. In such settings, it is natural to not want to give away more information than is necessary. To address this, we use the concept of privacy with respect to a function, ensuring that only information about the target function is available to all the parties, and no other information. We define a measure of privacy (essentially how close we are to this condition being satisfied) and show it satisfies a set of naturally desirable properties of such a measure. Using this privacy measure, we identify and construct entangled resource states that ensure privacy for a given function under different resource distributions and encoding dynamics, characterized by Hamiltonian evolution. For separable and parallel Hamiltonians, we prove that the GHZ state is the only private state for certain linear functions, with the minimum amount of required resources, up to SLOCC. Recognizing the vulnerability of this state to particle loss, we create families of private states, that remain robust even against loss of qubits, by incorporating additional resources. We then extend our findings to different resource distribution scenarios and Hamiltonians, resulting in a comprehensive set of private and robust states for distributed quantum estimation. These results advance the understanding of privacy and robustness in multi-parameter quantum sensing.

Featured image: Distributed sensing scenario, consisting of $a)$ a network of quantum nodes, capable of distributing entangled states, where $b)$ each of the nodes holds their own sets of qubits $\mathcal{N}_\mu$, with size $n_\mu$, which are the resources for quantum sensing in this case, even though our approach does not constraint the definition of resource to that of only qubits. One can also think of quantum resources as multiple sampling of a qubit to the encoding process. Nonetheless, the scheme is always the same: each qubit is in a distributed quantum state, and then undergoes some local (for each node) quantum dynamics which are parametrized by a local parameter $\theta_\mu$. After each node encodes their own parameters, all parties perform a joint measurement, from which they build an estimator for a global function of their parameters.

Poster presentation: PRIVATE AND ROBUST STATES FOR DISTRIBUTED QUANTUM SENSING

Popular summary

Quantum metrology and quantum sensing have been longstanding areas of interest in quantum information research. In particular, they are known to help improve the classical bounds one is subject to when estimating physical parameters in experiments. This means that there could potentially exist a quantum sensor more capable of measuring small variations of physical quantities, better than what any classical sensor could do.

Given the recent developments of quantum networks, both over large distances and local-area types of networks, distributed quantum sensing has emerged as a promising area of research with potential to affect in the near term. Among the many use-cases, one can already find clock synchronization protocols, optical interferometry proposals, and even some preliminary work regarding gravity and dark-matter experiments.

The setup and formalism for distributed quantum sensing has already been constructed and solidified as quantum sensor networks, and, more recently, security concerns have been introduced alongside a new important concept – that of privacy. In the distributed quantum sensing scenario, we are interested in estimating a function of parameters that are spatially distributed. This means, each of the parties are working together to find a function of their parameters. In this context, privacy means that each party cannot access the parameters of others, yet they can still compute the target function involving all parties' parameters by employing an entangled quantum state.

In our work we construct a set of tools to analyze the requirements for the privacy to be verified, and we find a clear distinction between quantum states which hold this property. We provide statements linking the number of resources, and the dynamics of the sensing involved in the estimation procedure, and come up with a complete set of quantum states which are private, for each of these settings. We also find that this set of private states is distinct from the set of non-private states. This implies that a secure and private sensing protocol could potentially take advantage of private states.

The link between the dynamics and the private states also gives rise to a geometrical-structure of distributed information itself. This structure is given by orthotopes, a set of vectors in a $d$-dimensional linear space, where $d$ is the number of parties involved in the estimation. In particular, the set of private information and corresponding private states is uniquely defined by this structure. Moreover, we also find that among the private states, one can find those which saturate the extractable information, meaning given a target function in the set of private functions of the structure, a private state is the one that holds the most possible information about such function. This has an interesting consequence: fixing the amount of resources used, one can either be very sensitive to one target function, or less sensitive to a set of functions.

► BibTeX data

@article{Bugalho2025privaterobuststates,
  doi = {10.22331/q-2025-01-15-1596},
  url = {https://doi.org/10.22331/q-2025-01-15-1596},
  title = {Private and {R}obust {S}tates for {D}istributed {Q}uantum {S}ensing},
  author = {Bugalho, Lu{\'{i}}s and Hassani, Majid and Omar, Yasser and Markham, Damian},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1596},
  month = jan,
  year = {2025}
}

► References

[1] Vittorio Giovannetti, Seth Lloyd, and Lorenzo Maccone. Quantum metrology. Physical Review Letters, 96 (1): 13–16, 2006. ISSN 10797114. 10.1103/PhysRevLett.96.010401.
https://doi.org/10.1103/PhysRevLett.96.010401

[2] Jasminder S. Sidhu and Pieter Kok. A Geometric Perspective on Quantum Parameter Estimation. AVS Quantum Science, 2 (1): 014701, July 2019. ISSN 2639-0213. 10.1116/1.5119961.
https://doi.org/10.1116/1.5119961

[3] C. L. Degen, F. Reinhard, and P. Cappellaro. Quantum sensing. Reviews of Modern Physics, 89 (3): 035002, July 2017. 10.1103/RevModPhys.89.035002.
https://doi.org/10.1103/RevModPhys.89.035002

[4] Zhibo Hou, Yan Jin, Hongzhen Chen, Jun-Feng Tang, Chang-Jiang Huang, Haidong Yuan, Guo-Yong Xiang, Chuan-Feng Li, and Guang-Can Guo. "Super-Heisenberg" and Heisenberg Scalings Achieved Simultaneously in the Estimation of a Rotating Field. Physical Review Letters, 126 (7): 70503, 2021. ISSN 1079-7114. 10.1103/PhysRevLett.126.070503.
https://doi.org/10.1103/PhysRevLett.126.070503

[5] Li-Zheng Liu, Yu-Zhe Zhang, Zheng-Da Li, Rui Zhang, Xu-Fei Yin, Yue-Yang Fei, Li Li, Nai-Le Liu, Feihu Xu, Yu-Ao Chen, and Jian-Wei Pan. Distributed quantum phase estimation with entangled photons. Nature Photonics, 15 (2): 137–142, February 2021. ISSN 1749-4885, 1749-4893. 10.1038/s41566-020-00718-2.
https://doi.org/10.1038/s41566-020-00718-2

[6] Marco Barbieri. Optical Quantum Metrology. PRX Quantum, 3 (1): 010202, January 2022. ISSN 2691-3399. 10.1103/PRXQuantum.3.010202.
https://doi.org/10.1103/PRXQuantum.3.010202

[7] Changhao Li, Mo Chen, and Paola Cappellaro. A geometric perspective: Experimental evaluation of the quantum Cramer-Rao bound, April 2022, arXiv:2204.13777. 10.48550/arXiv.2204.13777.
https://doi.org/10.48550/arXiv.2204.13777
arXiv:2204.13777

[8] Matteo G.A. Paris. Quantum estimation for quantum technology. International Journal of Quantum Information, 7 (SUPPL.): 125–137, 2009. ISSN 02197499. 10.1142/S0219749909004839.
https://doi.org/10.1142/S0219749909004839

[9] Stephanie Wehner, David Elkouss, and Ronald Hanson. Quantum internet: A vision for the road ahead. Science, 362 (6412): eaam9288, October 2018. ISSN 10959203. 10.1126/science.aam9288.
https://doi.org/10.1126/science.aam9288

[10] T. J. Proctor, P. A. Knott, and J. A. Dunningham. Networked quantum sensing, February 2017, arXiv:1702.04271. 10.48550/arXiv.1702.04271.
https://doi.org/10.48550/arXiv.1702.04271
arXiv:1702.04271

[11] Raja Yehia, Simon Neves, Eleni Diamanti, and Iordanis Kerenidis. Quantum City: Simulation of a practical near-term metropolitan quantum network, November 2022, arXiv:2211.01190. 10.48550/arXiv.2211.01190.
https://doi.org/10.48550/arXiv.2211.01190
arXiv:2211.01190

[12] Pavel Sekatski, Sabine Wölk, and Wolfgang Dür. Optimal distributed sensing in noisy environments. Physical Review Research, 2 (2): 1–8, May 2019. 10.1103/PhysRevResearch.2.023052.
https://doi.org/10.1103/PhysRevResearch.2.023052

[13] P. A. Knott, T. J. Proctor, A. J. Hayes, J. F. Ralph, P. Kok, and J. A. Dunningham. Local versus global strategies in multiparameter estimation. Physical Review A, 94 (6): 062312, December 2016. 10.1103/PhysRevA.94.062312.
https://doi.org/10.1103/PhysRevA.94.062312

[14] Zachary Eldredge, Michael Foss-Feig, Jonathan A. Gross, Steven L. Rolston, and Alexey V. Gorshkov. Optimal and secure measurement protocols for quantum sensor networks. Physical Review A, 97 (4): 042337, April 2018. ISSN 2469-9926. 10.1103/PhysRevA.97.042337.
https://doi.org/10.1103/PhysRevA.97.042337

[15] Timothy J. Proctor, Paul A. Knott, and Jacob A. Dunningham. Multiparameter Estimation in Networked Quantum Sensors. Physical Review Letters, 120 (8): 1–11, 2018. ISSN 10797114. 10.1103/PhysRevLett.120.080501.
https://doi.org/10.1103/PhysRevLett.120.080501

[16] Kevin Qian, Zachary Eldredge, Wenchao Ge, Guido Pagano, Christopher Monroe, J. V. Porto, and Alexey V. Gorshkov. Heisenberg-scaling measurement protocol for analytic functions with quantum sensor networks. Physical Review A, 100 (4): 042304, October 2019. ISSN 2469-9926. 10.1103/PhysRevA.100.042304.
https://doi.org/10.1103/PhysRevA.100.042304

[17] Zheshen Zhang and Quntao Zhuang. Distributed Quantum Sensing, October 2020, arXiv:2010.14744. 10.48550/arXiv.2010.14744.
https://doi.org/10.48550/arXiv.2010.14744
arXiv:2010.14744

[18] Timothy Qian, Jacob Bringewatt, Igor Boettcher, Przemyslaw Bienias, and Alexey V. Gorshkov. Optimal measurement of field properties with quantum sensor networks. Physical Review A, 103 (3): L030601, March 2021. ISSN 2469-9926. 10.1103/PhysRevA.103.L030601.
https://doi.org/10.1103/PhysRevA.103.L030601

[19] Huan Cao, Chao Zhang, Yun-Feng Huang, Bi-Heng Liu, Chuan-Feng Li, Philip Walther, and Guang-Can Guo. Experimental beating the standard quantum limit under non-markovian dephasing environment, August 2022, arXiv:2208.02543. 10.48550/arXiv.2208.02543.
https://doi.org/10.48550/arXiv.2208.02543
arXiv:2208.02543

[20] Peter Kómár, Eric M. Kessler, Michael Bishof, Liang Jiang, Anders S. Sørensen, Jun Ye, and Mikhail D. Lukin. A quantum network of clocks. Nature Physics, 10 (8): 582–587, October 2013. ISSN 1745-2473. 10.1038/nphys3000.
https://doi.org/10.1038/nphys3000

[21] B. C. Nichol, R. Srinivas, D. P. Nadlinger, P. Drmota, D. Main, G. Araneda, C. J. Ballance, and D. M. Lucas. An elementary quantum network of entangled optical atomic clocks. Nature, 609 (7928): 689–694, September 2022. ISSN 1476-4687. 10.1038/s41586-022-05088-z.
https://doi.org/10.1038/s41586-022-05088-z

[22] Emil T. Khabiboulline, Johannes Borregaard, Kristiaan De Greve, and Mikhail D. Lukin. Optical Interferometry with Quantum Networks. Physical Review Letters, 123 (7): 070504, September 2018. ISSN 0031-9007. 10.1103/PhysRevLett.123.070504.
https://doi.org/10.1103/PhysRevLett.123.070504

[23] T. Lévèque, C. Fallet, M. Mandea, R. Biancale, J. M. Lemoine, S. Tardivel, S. Delavault, A. Piquereau, S. Bourgogne, F. Pereira Dos Santos, B. Battelier, and Ph Bouyer. Gravity field mapping using laser-coupled quantum accelerometers in space. Journal of Geodesy, 95 (1): 1–19, 2021. ISSN 14321394. 10.1007/s00190-020-01462-9.
https://doi.org/10.1007/s00190-020-01462-9

[24] Lorcán O. Conlon, Thibault Michel, Giovanni Guccione, Kirk McKenzie, Syed M. Assad, and Ping Koy Lam. Enhancing the precision limits of interferometric satellite geodesy missions. npj Microgravity, 8 (1): 1–10, June 2022. ISSN 2373-8065. 10.1038/s41526-022-00204-9.
https://doi.org/10.1038/s41526-022-00204-9

[25] Anthony J. Brady, Xin Chen, Kewen Xiao, Yi Xia, Zhen Liu, Roni Harnik, Dalziel J. Wilson, Zheshen Zhang, and Quntao Zhuang. Entanglement-enhanced optomechanical sensor array for dark matter searches. Commun Phys, 6(237), 2023. https://doi.org/10.1038/s42005-023-01357-z.
https://doi.org/10.1038/s42005-023-01357-z

[26] Ivan Alonso et al. Cold Atoms in Space: Community Workshop Summary and Proposed Road-Map. EPJ Quantum Technology, 9 (1): 30, December 2022. ISSN 2662-4400, 2196-0763. https://doi.org/10.1140/epjqt/s40507-022-00147-w.
https://doi.org/10.1140/epjqt/s40507-022-00147-w

[27] B Fang, I Dutta, P Gillot, D Savoie, J Lautier, B Cheng, C L Garrido Alzar, R Geiger, S Merlet, F Pereira Dos Santos, and A Landragin. Metrology with Atom Interferometry: Inertial Sensors from Laboratory to Field Applications. Journal of Physics: Conference Series, 723: 012049, June 2016. ISSN 1742-6588, 1742-6596. 10.1088/1742-6596/723/1/012049.
https://doi.org/10.1088/1742-6596/723/1/012049

[28] Graham P. Greve, Chengyi Luo, Baochen Wu, and James K. Thompson. Entanglement-enhanced matter-wave interferometry in a high-finesse cavity. Nature, 610 (7932): 472–477, October 2022. ISSN 1476-4687. 10.1038/s41586-022-05197-9.
https://doi.org/10.1038/s41586-022-05197-9

[29] Bálint Koczor, Suguru Endo, Tyson Jones, Yuichiro Matsuzaki, and Simon C. Benjamin. Variational-state quantum metrology. New Journal of Physics, 22 (8): 1–14, August 2020. ISSN 13672630. 10.1088/1367-2630/ab965e.
https://doi.org/10.1088/1367-2630/ab965e

[30] Johannes Jakob Meyer, Johannes Borregaard, and Jens Eisert. A variational toolbox for quantum multi-parameter estimation. npj Quantum Information, 7 (1): 89, December 2021. ISSN 20566387. 10.1038/s41534-021-00425-y.
https://doi.org/10.1038/s41534-021-00425-y

[31] Raphael Kaubruegger, Athreya Shankar, Denis V. Vasilyev, and Peter Zoller. Optimal and Variational Multi-Parameter Quantum Metrology and Vector Field Sensing. PRX Quantum, 4 (020333), June 2023. 10.1103/PRXQuantum.4.020333.
https://doi.org/10.1103/PRXQuantum.4.020333

[32] Shuhong Hao, Haowei Shi, Christos N. Gagatsos, Mayank Mishra, Boulat Bash, Ivan Djordjevic, Saikat Guha, Quntao Zhuang, and Zheshen Zhang. Demonstration of Entanglement-Enhanced Covert Sensing. Physical Review Letters, 129 (1): 010501, June 2022. 10.1103/PhysRevLett.129.010501.
https://doi.org/10.1103/PhysRevLett.129.010501

[33] Hiroto Kasai, Yuki Takeuchi, Hideaki Hakoshima, Yuichiro Matsuzaki, and Yasuhiro Tokura. Anonymous Quantum Sensing. Journal of the Physical Society of Japan, 91 (7): 074005, July 2022. ISSN 0031-9015. 10.7566/JPSJ.91.074005.
https://doi.org/10.7566/JPSJ.91.074005

[34] Nathan Shettell, William J. Munro, Damian Markham, and Kae Nemoto. Practical limits of error correction for quantum metrology. New Journal of Physics, 23 (4): 043038, April 2021. ISSN 1367-2630. 10.1088/1367-2630/abf533.
https://doi.org/10.1088/1367-2630/abf533

[35] Kaoru Yamamoto, Suguru Endo, Hideaki Hakoshima, Yuichiro Matsuzaki, and Yuuki Tokunaga. Error-Mitigated Quantum Metrology via Virtual Purification. Physical Review Letters, 129 (25): 250503, December 2022. 10.1103/PhysRevLett.129.250503.
https://doi.org/10.1103/PhysRevLett.129.250503

[36] Jing Liu and Haidong Yuan. Control-enhanced multiparameter quantum estimation. Physical Review A, 96 (4), 2017. ISSN 24699934. 10.1103/PhysRevA.96.042114.
https://doi.org/10.1103/PhysRevA.96.042114

[37] Jesús Rubio and Jacob Dunningham. Bayesian multiparameter quantum metrology with limited data. Physical Review A, 101 (3), 2020. ISSN 24699934. 10.1103/PhysRevA.101.032114.
https://doi.org/10.1103/PhysRevA.101.032114

[38] Jesús Rubio, Paul A. Knott, Timothy J. Proctor, and Jacob A. Dunningham. Quantum sensing networks for the estimation of linear functions. Journal of Physics A: Mathematical and Theoretical, 53 (34): 344001, March 2020. ISSN 1751-8113. 10.1088/1751-8121/ab9d46.
https://doi.org/10.1088/1751-8121/ab9d46

[39] Jacob Bringewatt, Igor Boettcher, Pradeep Niroula, Przemyslaw Bienias, and Alexey V. Gorshkov. Protocols for estimating multiple functions with quantum sensor networks: Geometry and performance. Physical Review Research, 3 (3): 033011, July 2021. 10.1103/PhysRevResearch.3.033011.
https://doi.org/10.1103/PhysRevResearch.3.033011

[40] Nathan Shettell, Elham Kashefi, and Damian Markham. Cryptographic approach to quantum metrology. Physical Review A, 105 (1): L010401, January 2022a. 10.1103/PhysRevA.105.L010401.
https://doi.org/10.1103/PhysRevA.105.L010401

[41] Sean William Moore and Jacob Andrew Dunningham. Secure Quantum Remote Sensing Without Entanglement, February 2023, arXiv:2302.03617. 10.48550/arXiv.2302.03617.
https://doi.org/10.48550/arXiv.2302.03617
arXiv:2302.03617

[42] Nathan Shettell, Majid Hassani, and Damian Markham. Private network parameter estimation with quantum sensors, July 2022b, arXiv:2207.14450. 10.48550/arXiv.2207.14450.
https://doi.org/10.48550/arXiv.2207.14450
arXiv:2207.14450

[43] Nathan Shettell and Damian Markham. Graph States as a Resource for Quantum Metrology. Physical Review Letters, 124 (11): 110502, March 2020. ISSN 0031-9007. 10.1103/PhysRevLett.124.110502.
https://doi.org/10.1103/PhysRevLett.124.110502

[44] Jing Liu, Haidong Yuan, Xiao-Ming Lu, and Xiaoguang Wang. Quantum Fisher information matrix and multiparameter estimation. Journal of Physics A: Mathematical and Theoretical, 53 (2): 023001, December 2019. ISSN 1751-8121. 10.1088/1751-8121/ab5d4d.
https://doi.org/10.1088/1751-8121/ab5d4d

[45] Aaron Z. Goldberg, Luis L. Sánchez-Soto, and Hugo Ferretti. Intrinsic Sensitivity Limits for Multiparameter Quantum Metrology. Physical Review Letters, 127 (11): 110501, September 2021. 10.1103/PhysRevLett.127.110501.
https://doi.org/10.1103/PhysRevLett.127.110501

[46] Jarrod T. Reilly, John Drew Wilson, Simon B. Jäger, Christopher Wilson, and Murray J. Holland. Optimal Generators for Quantum Sensing. Physical Review Letters, 131 (150802), October 2023. 2010. 10.1103/PhysRevLett.131.150802.
https://doi.org/10.1103/PhysRevLett.131.150802

[47] Marcin Zwierz, Carlos A. Pérez-Delgado, and Pieter Kok. General optimality of the Heisenberg limit for quantum metrology. Physical Review Letters, 105 (18): 1–4, 2010. ISSN 00319007. 10.1103/PhysRevLett.105.180402.
https://doi.org/10.1103/PhysRevLett.105.180402

[48] M.A. Nielsen and I.L. Chuang. Quantum Computation and Quantum Information, volume 52. 2011. ISBN 978-1-107-00217-3. 10.1080/00107514.2011.587535.
https://doi.org/10.1080/00107514.2011.587535

[49] Shettell, Nathan and Markham, Damian Quantum Metrology with Delegated Tasks. Physical Review A, 106 (5): 052427, 2022. 10.1103/PhysRevA.106.052427.
https://doi.org/10.1103/PhysRevA.106.052427

[50] Sammy Ragy, Marcin Jarzyna, and Rafał Demkowicz-Dobrzański. Compatibility in multiparameter quantum metrology. Physical Review A, 94 (5): 052108, November 2016. ISSN 2469-9926, 2469-9934. 10.1103/PhysRevA.94.052108.
https://doi.org/10.1103/PhysRevA.94.052108

[51] Anupama Unnikrishnan and Damian Markham. Verification of graph states in an untrusted network. Physical Review A, 105 (5): 052420, May 2022. 10.1103/PhysRevA.105.052420.
https://doi.org/10.1103/PhysRevA.105.052420

[52] M. Hein, J. Eisert, and H. J. Briegel. Multiparty entanglement in graph states. Physical Review A - Atomic, Molecular, and Optical Physics, 69 (6): 1–22, 2004. ISSN 10502947. 10.1103/PhysRevA.69.062311.
https://doi.org/10.1103/PhysRevA.69.062311

[53] M. Hein, W. Dür, J. Eisert, R. Raussendorf, M. Van den Nest, and H. J. Briegel. Entanglement in Graph States and its Applications. Proceedings of the International School of Physics "Enrico Fermi", 162: 115–218, February 2006. ISSN 0074784X. 10.3254/978-1-61499-018-5-115.
https://doi.org/10.3254/978-1-61499-018-5-115

[54] Scott Aaronson and Daniel Gottesman. Improved simulation of stabilizer circuits. Physical Review A - Atomic, Molecular, and Optical Physics, 70 (5 A): 1–14, 2004. ISSN 10502947. 10.1103/PhysRevA.70.052328.
https://doi.org/10.1103/PhysRevA.70.052328

[55] Nathan Shettell. Quantum Information Techniques for Quantum Metrology. PhD thesis, 2022. 10.48550/arXiv.2201.01523.
https://doi.org/10.48550/arXiv.2201.01523

[56] Hong Tao and Xiaoqing Tan Quantum multiparameter estimation with graph states, June 2023, arXiv:2306.02518. 10.48550/arXiv.2306.02518.
https://doi.org/10.48550/arXiv.2306.02518
arXiv:2306.02518

[57] Rafał Demkowicz-Dobrzański, Jan Czajkowski, and Pavel Sekatski. Adaptive Quantum Metrology under General Markovian Noise. Physical Review X, 7 (4): 041009, October 2017. ISSN 2160-3308. 10.1103/PhysRevX.7.041009.
https://doi.org/10.1103/PhysRevX.7.041009

[58] Yingkai Ouyang, Nathan Shettell, and Damian Markham. Robust quantum metrology with explicit symmetric states. IEEE Transactions on Information Theory, 68 (3): 1809–1821, March 2022. ISSN 0018-9448, 1557-9654. 10.1109/TIT.2021.3132634.
https://doi.org/10.1109/TIT.2021.3132634

[59] Masahito Hayashi, Zi-Wen Liu, and Haidong Yuan. Global Heisenberg scaling in noisy and practical phase estimation. Quantum Science and Technology, 7 (2): 025030, April 2022. ISSN 2058-9565. 10.1088/2058-9565/ac5d7e.
https://doi.org/10.1088/2058-9565/ac5d7e

[60] A. T. Rezakhani, M. Hassani, and S. Alipour. Continuity of the quantum Fisher information. Physical Review A, 100 (3): 32317, 2019. ISSN 24699934. 10.1103/PhysRevA.100.032317.
https://doi.org/10.1103/PhysRevA.100.032317

[61] Luís Bugalho, Bruno C. Coutinho, Francisco A. Monteiro, and Yasser Omar. Distributing Multipartite Entanglement over Noisy Quantum Networks. Quantum, 7: 920, February 2023. 10.22331/q-2023-02-09-920.
https://doi.org/10.22331/q-2023-02-09-920

[62] Reinhard F. Werner. Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model. Physical Review A, 40 (8): 4277–4281, 1989. ISSN 10502947. 10.1103/PhysRevA.40.4277.
https://doi.org/10.1103/PhysRevA.40.4277

[63] Raffaele Santagati, Jianwei Wang, Antonio A. Gentile, Stefano Paesani, Nathan Wiebe, Jarrod R. McClean, Sam Morley-Short, Peter J. Shadbolt, Damien Bonneau, Joshua W. Silverstone, David P. Tew, Xiaoqi Zhou, Jeremy L. O'Brien, and Mark G. Thompson. Witnessing eigenstates for quantum simulation of Hamiltonian spectra. Science Advances, 4 (1): eaap9646, January 2018. 10.1126/sciadv.aap9646.
https://doi.org/10.1126/sciadv.aap9646

[64] Richard Meister and Simon C. Benjamin. Resource-frugal Hamiltonian eigenstate preparation via repeated quantum phase estimation measurements, December 2022, arXiv:2212.00846. 10.48550/arXiv.2212.00846.
https://doi.org/10.48550/arXiv.2212.00846
arXiv:2212.00846

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[4] Faramarz Farahi and Jose Luis Santos, "Sensors: The Building Blocks of a Technology-Driven Future", IEEE Sensors Reviews 3, 48 (2026).

[5] Nicolas Laurent-Puig, Laura dos Santos Martins, Matilde Baroni, Santiago Schneider, Luis Bugalho, Majid Hassani, Sean Moore, Damian Markham, and Eleni Diamanti, 2026 International Conference on Quantum Communications, Networking, and Computing (QCNC) 513 (2026) ISBN:979-8-3315-6110-9.

[6] Tzula B. Propp, Jeroen Grimbergen, Emil R. Hellebek, Junior R. Gonzales-Ureta, Janice van Dam, Joshua A. Slater, Anders S. Sørensen, and Stephanie D. C. Wehner, "Quantum strategies to overcome classical multiplexing limits", Physical Review A 114 1, 012603 (2026).

[7] Jia-Xuan Liu, Hai-Long Shi, Chunfeng Wu, and Sixia Yu, "Quantum estimation with state symmetry-induced optimal measurements", Nature Communications 17 1, 7898 (2026).

[8] Jarn de Jong, Santiago Scheiner, Naomi R. Solomons, Ziad Chaoui, Damian Markham, and Anna Pappa, "Anonymous and private parameter estimation in networks of quantum sensors", Physical Review Applied 24 5, 054053 (2025).

[9] Ufuk Keskin, Stefano Marano, and Moe Z. Win, 2026 International Conference on Quantum Communications, Networking, and Computing (QCNC) 159 (2026) ISBN:979-8-3315-6110-9.

[10] J Knörzer, X Liu, B F Schiffer, and J Tura, "Distributed quantum information processing: a review of recent progress", Reports on Progress in Physics 89 7, 074401 (2026).

[11] Himanshu Sahu and Hari Prabhat Gupta, "BTS-QSN: Budgeting and Task Scheduling in Quantum Sensor Networks", IEEE Sensors Journal 26 15, 23466 (2026).

[12] Julia Freund, Alexander Pirker, Lina Vandré, and Wolfgang Dür, "Graph state extraction from two-dimensional cluster states", New Journal of Physics 27 9, 094505 (2025).

[13] Gabriele Bizzarri, Marco Barbieri, Mylenne Manrique, Miranda Parisi, Fabio Bruni, Ilaria Gianani, and Matteo Rosati, "Faithful and secure distributed quantum sensing under general-coherent attacks", npj Quantum Information 12 1, 33 (2026).

[14] Farzad Kianvash, Marco Barbieri, and Matteo Rosati, "Private Remote Phase Estimation over a Lossy Quantum Channel", Physical Review Letters 136 6, 060805 (2026).

[15] Joseph Ho, Jonathan W Webb, Russell M J Brooks, Federico Grasselli, Erik Gauger, and Alessandro Fedrizzi, "Quantum-private distributed sensing", Journal of Physics: Photonics 8 2, 025006 (2026).

[16] J. Bate, A. Hamann, M. Canteri, A. Winkler, Z. X. Koong, V. Krutyanskiy, W. Dür, and B. P. Lanyon, "Experimental Distributed Quantum Sensing in a Noisy Environment", Physical Review Letters 135 22, 220801 (2025).

[17] Arne Hamann, Paul Aigner, Wolfgang Dür, and Pavel Sekatski, "Selective and noise-resilient wave estimation with quantum sensor networks", Quantum Science and Technology 10 3, 035028 (2025).

[18] Luís Bugalho, Yasser Omar, and Damian Markham, "A Framework for Spatial Quantum Sensing", arXiv:2602.12193, (2026).

[19] Daniel J. Spencer, Kaiyan Shi, Emil T. Khabiboulline, Gorjan Alagic, and Alexey V. Gorshkov, "Differentially private quantum sensor networks", arXiv:2607.06521, (2026).

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