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Experimental Distributed Quantum Sensing in a Noisy Environment

J. Bate1, A. Hamann2, M. Canteri1, A. Winkler1, Z. X. Koong1, V. Krutyanskiy1, W. Dür2, and B. P. Lanyon1,*

  • *Contact author: ben.lanyon@uibk.ac.at
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Phys. Rev. Lett. 135, 220801 – Published 24 November, 2025

DOI: https://doi.org/10.1103/3hgx-wcdn

Abstract

The precision advantages offered by harnessing the quantum states of sensors can be readily compromised by noise. However, when the noise has a different spatial function than the signal of interest, recent theoretical work shows how the advantage can be maintained and even significantly improved. In this Letter, we experimentally demonstrate the associated sensing protocol, using trapped-ion sensors. An entangled state of multidimensional sensors is created that isolates and optimally detects a signal, while being insensitive to otherwise overwhelming noise fields with different spatial profiles over the sensor locations. The quantum protocol is found to outperform a perfect implementation of the best comparable strategy without sensor entanglement. While our demonstration is carried out for magnetic and electromagnetic fields over a few microns, the technique is readily applicable over arbitrary distances and for arbitrary fields, thus present a promising application for emerging quantum sensor networks.

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References (83)

  1. C. M. Caves, Quantum-mechanical noise in an interferometer, Phys. Rev. D 23, 1693 (1981).
  2. S. L. Braunstein and C. M. Caves, Statistical distance and the geometry of quantum states, Phys. Rev. Lett. 72, 3439 (1994).
  3. M. Kitagawa and M. Ueda, Squeezed spin states, Phys. Rev. A 47, 5138 (1993).
  4. J. J.. Bollinger, W. M. Itano, D. J. Wineland, and D. J. Heinzen, Optimal frequency measurements with maximally correlated states, Phys. Rev. A 54, R4649 (1996).
  5. V. Giovannetti, S. Lloyd, and L. Maccone, Advances in quantum metrology, Nat. Photonics 5, 222 (2011).
  6. L. Pezzè, A. Smerzi, M. K. Oberthaler, R. Schmied, and P. Treutlein, Quantum metrology with nonclassical states of atomic ensembles, Rev. Mod. Phys. 90, 035005 (2018).
  7. R. Demkowicz-Dobrzański, M. Jarzyna, and J. Kołodyński, Chapter Four—Quantum Limits in Optical Interferometry (Elsevier, New York, 2015), pp. 345–435.
  8. C. F. Roos, M. Chwalla, K. Kim, M. Riebe, and R. Blatt, ‘Designer atoms’ for quantum metrology, Nature (London) 443, 316 (2006).
  9. K. A. Gilmore, M. Affolter, R. J. Lewis-Swan, D. Barberena, E. Jordan, A. M. Rey, and J. J. Bollinger, Quantum-enhanced sensing of displacements and electric fields with two-dimensional trapped-ion crystals, Science 373, 673 (2021).
  10. J. M. Taylor, P. Cappellaro, L. Childress, L. Jiang, D. Budker, P. R. Hemmer, A. Yacoby, R. Walsworth, and M. D. Lukin, High-sensitivity diamond magnetometer with nanoscale resolution, Nat. Phys. 4, 810 (2008).
  11. LIGO Scientific Collaboration, Enhanced sensitivity of the LIGO gravitational wave detector by using squeezed states of light, Nat. Photonics 7, 613 (2013).
  12. C. Salducci, Y. Bidel, M. Cadoret, S. Darmon, N. Zahzam, A. Bonnin, S. Schwartz, C. Blanchard, and A. Bresson, Quantum sensing of acceleration and rotation by interfering magnetically launched atoms, Sci. Adv. 10, eadq4498 (2024).
  13. C. Hempel, B. P. Lanyon, P. Jurcevic, R. Gerritsma, R. Blatt, and C. F. Roos, Entanglement-enhanced detection of single-photon scattering events, Nat. Photonics 7, 630 (2013).
  14. F. Wolf, C. Shi, J. C. Heip, M. Gessner, L. Pezzè, A. Smerzi, M. Schulte, K. Hammerer, and P. O. Schmidt, Motional fock states for quantum-enhanced amplitude and phase measurements with trapped ions, Nat. Commun. 10, 2929 (2019).
  15. T. J. Proctor, P. A. Knott, and J. A. Dunningham, Multiparameter estimation in networked quantum sensors, Phys. Rev. Lett. 120, 080501 (2018).
  16. Z. Eldredge, M. Foss-Feig, J. A. Gross, S. L. Rolston, and A. V. Gorshkov, Optimal and secure measurement protocols for quantum sensor networks, Phys. Rev. A 97, 042337 (2018).
  17. K. Qian, Z. Eldredge, W. Ge, G. Pagano, C. Monroe, J. V. Porto, and A. V. Gorshkov, Heisenberg-scaling measurement protocol for analytic functions with quantum sensor networks, Phys. Rev. A 100, 042304 (2019).
  18. N. Shettell and D. Markham, Graph states as a resource for quantum metrology, Phys. Rev. Lett. 124, 110502 (2020).
  19. J. Rubio, P. A. Knott, T. J. Proctor, and J. A. Dunningham, Quantum sensing networks for the estimation of linear functions, J. Phys. A 53, 344001 (2020).
  20. J. Bringewatt, I. Boettcher, P. Niroula, P. Bienias, and A. V. Gorshkov, Protocols for estimating multiple functions with quantum sensor networks: Geometry and performance, Phys. Rev. Res. 3, 033011 (2021).
  21. N. Shettell, M. Hassani, and D. Markham, Private network parameter estimation with quantum sensors, arXiv:2207.14450.
  22. L. Bugalho, M. Hassani, Y. Omar, and D. Markham, Private and robust states for distributed quantum sensing, Quantum 9, 1596 (2025).
  23. M. Hassani, S. Scheiner, M. G. A. Paris, and D. Markham, Privacy in networks of quantum sensors, Phys. Rev. Lett. 134, 030802 (2025).
  24. L.-Z. Liu, Y.-Z. Zhang, Z.-D. Li, R. Zhang, X.-F. Yin, Y.-Y. Fei, L. Li, N.-L. Liu, F. Xu, Y.-A. Chen, and J.-W. Pan, Distributed quantum phase estimation with entangled photons, Nat. Photonics 15, 137 (2021).
  25. D.-H. Kim, S. Hong, Y.-S. Kim, Y. Kim, S.-W. Lee, R. C. Pooser, K. Oh, S.-Y. Lee, C. Lee, and H.-T. Lim, Distributed quantum sensing of multiple phases with fewer photons, Nat. Commun. 15, 266 (2024).
  26. S.-R. Zhao, Y.-Z. Zhang, W.-Z. Liu, J.-Y. Guan, W. Zhang, C.-L. Li, B. Bai, M.-H. Li, Y. Liu, L. You, J. Zhang, J. Fan, F. Xu, Q. Zhang, and J.-W. Pan, Field demonstration of distributed quantum sensing without post-selection, Phys. Rev. X 11, 031009 (2021).
  27. 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 (London) 609, 689 (2022).
  28. H. Hainzer, D. Kiesenhofer, T. Ollikainen, M. Bock, F. Kranzl, M. K. Joshi, G. Yoeli, R. Blatt, T. Gefen, and C. F. Roos, Correlation spectroscopy with multiqubit-enhanced phase estimation, Phys. Rev. X 14, 011033 (2022).
  29. B. K. Malia, Y. Wu, J. Martínez-Rincón, and M. A. Kasevich, Distributed quantum sensing with mode-entangled spin-squeezed atomic states, Nature (London) 612, 661 (2022).
  30. R. Demkowicz-Dobrzański, J. Czajkowski, and P. Sekatski, Adaptive quantum metrology under general Markovian noise, Phys. Rev. X 7, 041009 (2017).
  31. S. Zhou, M. Zhang, J. Preskill, and L. Jiang, Achieving the Heisenberg limit in quantum metrology using quantum error correction, Nat. Commun. 9, 78 (2018).
  32. T. Monz, P. Schindler, J. T. Barreiro, M. Chwalla, D. Nigg, W. A. Coish, M. Harlander, W. Hänsel, M. Hennrich, and R. Blatt, 14-qubit entanglement: Creation and coherence, Phys. Rev. Lett. 106, 130506 (2011).
  33. A. Omran, H. Levine, A. Keesling, G. Semeghini, T. T. Wang, S. Ebadi, H. Bernien, A. S. Zibrov, H. Pichler, S. Choi, J. Cui, M. Rossignolo, P. Rembold, S. Montangero, T. Calarco, M. Endres, M. Greiner, V. Vuletić, and M. D. Lukin, Generation and manipulation of Schrödinger cat states in Rydberg atom arrays, Science 365, 570 (2019).
  34. Z. Bao, S. Xu, Z. Song, K. Wang, L. Xiang, Z. Zhu, J. Chen, F. Jin, X. Zhu, Y. Gao et al., Creating and controlling global Greenberger-Horne-Zeilinger entanglement on quantum processors, Nat. Commun. 15, 8823 (2024).
  35. E. M. Kessler, I. Lovchinsky, A. O. Sushkov, and M. D. Lukin, Quantum error correction for metrology, Phys. Rev. Lett. 112, 150802 (2014).
  36. G. Arrad, Y. Vinkler, D. Aharonov, and A. Retzker, Increasing sensing resolution with error correction, Phys. Rev. Lett. 112, 150801 (2014).
  37. P. Sekatski, M. Skotiniotis, J. Kołodyński, and W. Dür, Quantum metrology with full and fast quantum control, Quantum 1, 27 (2017).
  38. P. Faist, M. P. Woods, V. V. Albert, J. M. Renes, J. Eisert, and J. Preskill, Time-energy uncertainty relation for noisy quantum metrology, PRX Quantum 4, 040336 (2023).
  39. P. Sekatski, S. Wölk, and W. Dür, Optimal distributed sensing in noisy environments, Phys. Rev. Res. 2, 023052 (2020).
  40. C. W. Helstrom, Quantum detection and estimation theory, J. Stat. Phys. 1, 231 (1969).
  41. P. Zanardi and M. Rasetti, Noiseless quantum codes, Phys. Rev. Lett. 79, 3306 (1997).
  42. A. Hamann, P. Sekatski, and W. Dür, Optimal distributed multi-parameter estimation in noisy environments, Quantum Sci. Technol. 9, 035005 (2024).
  43. S. Wölk, P. Sekatski, and W. Dür, Noisy distributed sensing in the Bayesian regime, Quantum Sci. Technol. 5, 045003 (2020).
  44. A. Hamann, P. Sekatski, and W. Dür, Approximate decoherence free subspaces for distributed sensing, Quantum Sci. Technol. 7, 025003 (2022).
  45. M. Landini, M. Fattori, L. Pezzè, and A. Smerzi, Phase-noise protection in quantum-enhanced differential interferometry, New J. Phys. 16, 113074 (2014).
  46. See Supplemental Material, which includes Refs. [5,32,39,40,47–59], at http://link.aps.org/supplemental/10.1103/3hgx-wcdn for details on the estimation protocols, experimental methods, and supporting experimental and theoretical results.
  47. C. R. Rao, Information and the accuracy attainable in the estimation of statistical parameters, in Breakthroughs in Statistics: Foundations and Basic Theory, edited by S. Kotz and N. L. Johnson (Springer, New York, 1992), pp. 235–247.
  48. H. Cramér, Mathematical Methods of Statistics, 1st ed., Princeton Mathematical Series (Princeton University Press, Princeton, New Jersey, 1946).
  49. R. A. Fisher and E. J. Russell, On the mathematical foundations of theoretical statistics, Phil. Trans. R. Soc. A 222, 309 (1922).
  50. M. G. A. Paris, Quantum estimation for quantum technology, Int. J. Quantum. Inform. 07, 125 (2009).
  51. M. Canteri, Single-atom-focused laser for photon generation and qubit control, Master’s thesis, University of Innsbruck, 2020.
  52. D. F. V. James, Quantum dynamics of cold trapped ions with application to quantum computation, Appl. Phys. B 66, 181 (1998).
  53. C. Roos, Controlling the quantum state of trapped ions, Ph.D. thesis, University of Innsbruck, 2000.
  54. M. Ringbauer, M. Meth, L. Postler, R. Stricker, R. Blatt, P. Schindler, and T. Monz, A universal qudit quantum processor with trapped ions, Nat. Phys. 18, 1053 (2022).
  55. M. G. Bertsch, Optical clocks with trapped ions, Ph.D. thesis, University of Innsbruck, 2023.
  56. S. Chessa and V. Giovannetti, Quantum capacity analysis of multi-level amplitude damping channels, Commun. Phys. 4, 22 (2021).
  57. M. M. Wilde, Quantum Information Theory (Cambridge University Press, Cambridge, England, 2013), p. 138.
  58. A. Sørensen and K. Mølmer, Entanglement and quantum computation with ions in thermal motion, Phys. Rev. A 62, 022311 (2000).
  59. G. Kirchmair, J. Benhelm, F. Zähringer, R. Gerritsma, C. F. Roos, and R. Blatt, Deterministic entanglement of ions in thermal states of motion, New J. Phys. 11, 023002 (2009).
  60. D. M. Greenberger, M. A. Horne, and A. Zeilinger, Going beyond Bell’s theorem, in Bell’s Theorem, Quantum Theory and Conceptions of the Universe (Springer, Dordrecht, Netherlands, 1989), pp. 69–72.
  61. K. Mølmer and A. Sørensen, Multiparticle entanglement of hot trapped ions, Phys. Rev. Lett. 82, 1835 (1999).
  62. T. R. Tan, J. P. Gaebler, R. Bowler, Y. Lin, J. D. Jost, D. Leibfried, and D. J. Wineland, Demonstration of a dressed-state phase gate for trapped ions, Phys. Rev. Lett. 110, 263002 (2013).
  63. E. R. Clements, M. E. Kim, K. Cui, A. M. Hankin, S. M. Brewer, J. Valencia, J.-S. Chen, C.-W. Chou, D. R. Leibrandt, and D. B. Hume, Lifetime-limited interrogation of two independent al27+ clocks using correlation spectroscopy, Phys. Rev. Lett. 125, 243602 (2020).
  64. O. Gühne and P. Hyllus, Investigating three qubit entanglement with local measurements, Int. J. Theor. Phys. 42, 1001 (2003).
  65. S. A. Moses, C. H. Baldwin, M. S. Allman, R. Ancona, L. Ascarrunz, C. Barnes, J. Bartolotta, B. Bjork, P. Blanchard, M. Bohn et al., A race-track trapped-ion quantum processor, Phys. Rev. X 13, 041052 (2023).
  66. C. D. Bruzewicz, J. Chiaverini, R. McConnell, and J. M. Sage, Trapped-ion quantum computing: Progress and challenges, Appl. Phys. Rev. 6, 021314 (2019).
  67. D. Bluvstein, H. Levine, G. Semeghini, T. T. Wang, S. Ebadi, M. Kalinowski, A. Keesling, N. Maskara, H. Pichler, M. Greiner, V. Vuletić, and M. D. Lukin, A quantum processor based on coherent transport of entangled atom arrays, Nature (London) 604, 451 (2022).
  68. M. Kjaergaard, M. E. Schwartz, J. Braumüller, P. Krantz, J. I.-J. Wang, S. Gustavsson, and W. D. Oliver, Superconducting qubits: Current state of play, Annu. Rev. Condens. Matter Phys. 11, 369 (2020).
  69. T. van Leent, M. Bock, F. Fertig, R. Garthoff, S. Eppelt, Y. Zhou, P. Malik, M. Seubert, T. Bauer, W. Rosenfeld, W. Zhang, C. Becher, and H. Weinfurter, Entangling single atoms over 33 km telecom fibre, Nature (London) 607, 69 (2022).
  70. M. Pompili, S. L. N. Hermans, S. Baier, H. K. C. Beukers, P. C. Humphreys, R. N. Schouten, R. F. L. Vermeulen, M. J. Tiggelman, L. dos Santos Martins, B. Dirkse, S. Wehner, and R. Hanson, Realization of a multinode quantum network of remote solid-state qubits, Science 372, 259 (2021).
  71. D. L. Moehring, P. Maunz, S. Olmschenk, K. C. Younge, D. N. Matsukevich, L.-M. Duan, and C. Monroe, Entanglement of single-atom quantum bits at a distance, Nature (London) 449, 68 (2007).
  72. L. J. Stephenson, D. P. Nadlinger, B. C. Nichol, S. An, P. Drmota, T. G. Ballance, K. Thirumalai, J. F. Goodwin, D. M. Lucas, and C. J. Ballance, High-rate, high-fidelity entanglement of qubits across an elementary quantum network, Phys. Rev. Lett. 124, 110501 (2020).
  73. S. Ritter, C. Nölleke, C. Hahn, A. Reiserer, A. Neuzner, M. Uphoff, M. Mücke, E. Figueroa, J. Bochmann, and G. Rempe, An elementary quantum network of single atoms in optical cavities, Nature (London) 484, 195 (2012).
  74. A. Delteil, Z. Sun, W.-b. Gao, E. Togan, S. Faelt, and A. Imamoğlu, Generation of heralded entanglement between distant hole spins, Nat. Phys. 12, 218 (2016).
  75. R. Stockill, M. J. Stanley, L. Huthmacher, E. Clarke, M. Hugues, A. J. Miller, C. Matthiesen, C. Le Gall, and M. Atatüre, Phase-tuned entangled state generation between distant spin qubits, Phys. Rev. Lett. 119, 010503 (2017).
  76. P. Magnard, S. Storz, P. Kurpiers, J. Schär, F. Marxer, J. Lütolf, T. Walter, J.-C. Besse, M. Gabureac, K. Reuer, A. Akin, B. Royer, A. Blais, and A. Wallraff, Microwave quantum link between superconducting circuits housed in spatially separated cryogenic systems, Phys. Rev. Lett. 125, 260502 (2020).
  77. J.-L. Liu, X.-Y. Luo, Y. Yu, C.-Y. Wang, B. Wang, Y. Hu, J. Li, M.-Y. Zheng, B. Yao, Z. Yan et al., Creation of memory–memory entanglement in a metropolitan quantum network, Nature (London) 629, 579 (2024).
  78. C. M. Knaut, A. Suleymanzade, Y.-C. Wei, D. R. Assumpcao, P.-J. Stas, Y. Q. Huan, B. Machielse, E. N. Knall, M. Sutula, G. Baranes, M. D. Lukin et al., Entanglement of nanophotonic quantum memory nodes in a telecom network, Nature (London) 629, 573 (2024).
  79. V. Krutyanskiy, M. Galli, V. Krcmarsky, S. Baier, D. A. Fioretto, Y. Pu, A. Mazloom, P. Sekatski, M. Canteri, M. Teller, J. Schupp, J. Bate, M. Meraner, N. Sangouard, B. P. Lanyon, and T. E. Northup, Entanglement of trapped-ion qubits separated by 230 meters, Phys. Rev. Lett. 130, 050803 (2023).
  80. V. Krutyanskiy, M. Canteri, M. Meraner, V. Krcmarsky, and B. P. Lanyon, Multimode ion-photon entanglement over 101 kilometers, PRX Quantum 5, 020308 (2024).
  81. V. Krutyanskiy, M. Canteri, M. Meraner, J. Bate, V. Krcmarsky, J. Schupp, N. Sangouard, and B. P. Lanyon, Telecom-wavelength quantum repeater node based on a trapped-ion processor, Phys. Rev. Lett. 130, 213601 (2023).
  82. P. Drmota, D. Main, D. P. Nadlinger, B. C. Nichol, M. A. Weber, E. M. Ainley, A. Agrawal, R. Srinivas, G. Araneda, C. J. Ballance, and D. M. Lucas, Robust quantum memory in a trapped-ion quantum network node, Phys. Rev. Lett. 130, 090803 (2023).
  83. J. Bate, A. Hamann, M. Canteri, A. Winkler, Z. X. Koong, V. Krutyanskiy, W. Dür, B. P. Lanyon, Experimental distributed quantum sensing in a noisy environment, 10.5281/zenodo.17425896 (2025).

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Phys. Rev. Lett. 135, 220801– Published 24 November, 2025
  • Received 10 January 2025
  • Revised 9 May 2025
  • Accepted 15 October 2025
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