Metrologically optimal quantum states under noise

Chao Yin; Victor V. Albert; Sisi Zhou

doi:10.22331/q-2026-04-01-2050

Metrologically optimal quantum states under noise

Chao Yin¹, Victor V. Albert², and Sisi Zhou^3,4

¹Department of Physics and Center for Theory of Quantum Matter, University of Colorado, Boulder CO 80309, USA
²Joint Center for Quantum Information and Computer Science, NIST/University of Maryland, College Park, Maryland, USA
³Perimeter Institute for Theoretical Physics, Waterloo, ON, N2L 2Y5, Canada
⁴Department of Physics and Astronomy, Department of Applied Mathematics, and Institute for Quantum Computing, University of Waterloo, Ontario N2L 2Y5, Canada

Published:	2026-04-01, volume 10, page 2050
Editor:	Dennis Rätzel
Eprint:	arXiv:2408.00079v4
Doi:	https://doi.org/10.22331/q-2026-04-01-2050
Citation:	Quantum 10, 2050 (2026).

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.

Abstract

We propose a class of metrological resource states whose quantum Fisher information scales optimally in both system size and noise rate. In these states, qubits are partitioned into sensing groups with relatively large correlations within a group but small correlations between groups. The states are obtainable from local Hamiltonian evolution, and we design a metrologically optimal and efficient measurement protocol utilizing time-reversed dynamics and single-qubit on-site measurements. Using quantum domino dynamics, we also present a protocol free of the time-reversal step that has an estimation error roughly twice the best possible value. Finally, we show that spin squeezed states are also optimal for noisy metrology under general conditions.

► BibTeX data

@article{Yin2026metrologically,
  doi = {10.22331/q-2026-04-01-2050},
  url = {https://doi.org/10.22331/q-2026-04-01-2050},
  title = {Metrologically optimal quantum states under noise},
  author = {Yin, Chao and Albert, Victor V. and Zhou, Sisi},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {10},
  pages = {2050},
  month = apr,
  year = {2026}
}

► References

[1] Vittorio Giovannetti, Seth Lloyd, and Lorenzo Maccone. ``Advances in quantum metrology''. Nature photonics 5, 222–229 (2011).
https://doi.org/10.1038/nphoton.2011.35

[2] C. L. Degen, F. Reinhard, and P. Cappellaro. ``Quantum sensing''. Rev. Mod. Phys. 89, 035002 (2017).
https://doi.org/10.1103/RevModPhys.89.035002

[3] Luca Pezzè, Augusto Smerzi, Markus K. Oberthaler, Roman Schmied, and Philipp Treutlein. ``Quantum metrology with nonclassical states of atomic ensembles''. Rev. Mod. Phys. 90, 035005 (2018).
https://doi.org/10.1103/RevModPhys.90.035005

[4] Stefano Pirandola, Bhaskar Roy Bardhan, Tobias Gehring, Christian Weedbrook, and Seth Lloyd. ``Advances in photonic quantum sensing''. Nat. Photonics. 12, 724 (2018).
https://doi.org/10.1038/s41566-018-0301-6

[5] Carlton M. Caves. ``Quantum-mechanical noise in an interferometer''. Phys. Rev. D 23, 1693–1708 (1981).
https://doi.org/10.1103/PhysRevD.23.1693

[6] Bernard Yurke, Samuel L. McCall, and John R. Klauder. ``Su(2) and su(1,1) interferometers''. Phys. Rev. A 33, 4033–4054 (1986).
https://doi.org/10.1103/PhysRevA.33.4033

[7] LIGO Collaboration. ``A gravitational wave observatory operating beyond the quantum shot-noise limit''. Nature Physics 7, 962–965 (2011).
https://doi.org/10.1038/nphys2083

[8] LIGO Collaboration. ``Enhanced sensitivity of the ligo gravitational wave detector by using squeezed states of light''. Nature Photonics 7, 613–619 (2013).
https://doi.org/10.1038/nphoton.2013.177

[9] David Le Sage, Koji Arai, David R Glenn, Stephen J DeVience, Linh M Pham, Lilah Rahn-Lee, Mikhail D Lukin, Amir Yacoby, Arash Komeili, and Ronald L Walsworth. ``Optical magnetic imaging of living cells''. Nature 496, 486–489 (2013).
https://doi.org/10.1038/nature12072

[10] Gabriela Barreto Lemos, Victoria Borish, Garrett D Cole, Sven Ramelow, Radek Lapkiewicz, and Anton Zeilinger. ``Quantum imaging with undetected photons''. Nature 512, 409–412 (2014).
https://doi.org/10.1038/nature13586

[11] Mankei Tsang, Ranjith Nair, and Xiao-Ming Lu. ``Quantum theory of superresolution for two incoherent optical point sources''. Physical Review X 6, 031033 (2016).
https://doi.org/10.1103/PhysRevX.6.031033

[12] MH Abobeih, J Randall, CE Bradley, HP Bartling, MA Bakker, MJ Degen, M Markham, DJ Twitchen, and TH Taminiau. ``Atomic-scale imaging of a 27-nuclear-spin cluster using a quantum sensor''. Nature 576, 411–415 (2019).
https://doi.org/10.1038/s41586-019-1834-7

[13] D. J. Wineland, J. J. Bollinger, W. M. Itano, F. L. Moore, and D. J. Heinzen. ``Spin squeezing and reduced quantum noise in spectroscopy''. Phys. Rev. A 46, R6797–R6800 (1992).
https://doi.org/10.1103/PhysRevA.46.R6797

[14] J. J. Bollinger, Wayne M. Itano, D. J. Wineland, and D. J. Heinzen. ``Optimal frequency measurements with maximally correlated states''. Phys. Rev. A 54, R4649–R4652 (1996).
https://doi.org/10.1103/PhysRevA.54.R4649

[15] D. Leibfried, M. D. Barrett, T. Schaetz, J. Britton, J. Chiaverini, W. M. Itano, J. D. Jost, C. Langer, and D. J. Wineland. ``Toward heisenberg-limited spectroscopy with multiparticle entangled states''. Science 304, 1476–1478 (2004).
https://doi.org/10.1126/science.1097576

[16] JM Taylor, Paola Cappellaro, L Childress, Liang Jiang, Dmitry Budker, PR Hemmer, Amir Yacoby, R Walsworth, and MD Lukin. ``High-sensitivity diamond magnetometer with nanoscale resolution''. Nature Physics 4, 810–816 (2008).
https://doi.org/10.1038/nphys1075

[17] Hengyun Zhou, Joonhee Choi, Soonwon Choi, Renate Landig, Alexander M Douglas, Junichi Isoya, Fedor Jelezko, Shinobu Onoda, Hitoshi Sumiya, Paola Cappellaro, et al. ``Quantum metrology with strongly interacting spin systems''. Physical review X 10, 031003 (2020).
https://doi.org/10.1103/PhysRevX.10.031003

[18] Till Rosenband, DB Hume, PO Schmidt, Chin-Wen Chou, Anders Brusch, Luca Lorini, WH Oskay, Robert E Drullinger, Tara M Fortier, Jason E Stalnaker, et al. ``Frequency ratio of al+ and hg+ single-ion optical clocks; metrology at the 17th decimal place''. Science 319, 1808–1812 (2008).
https://doi.org/10.1126/science.1154622

[19] Jürgen Appel, Patrick Joachim Windpassinger, Daniel Oblak, U Busk Hoff, Niels Kjærgaard, and Eugene Simon Polzik. ``Mesoscopic atomic entanglement for precision measurements beyond the standard quantum limit''. Proceedings of the National Academy of Sciences 106, 10960–10965 (2009).
https://doi.org/10.1073/pnas.0901550106

[20] Andrew D Ludlow, Martin M Boyd, Jun Ye, Ekkehard Peik, and Piet O Schmidt. ``Optical atomic clocks''. Reviews of Modern Physics 87, 637 (2015).
https://doi.org/10.1103/RevModPhys.87.637

[21] Raphael Kaubruegger, Denis V Vasilyev, Marius Schulte, Klemens Hammerer, and Peter Zoller. ``Quantum variational optimization of ramsey interferometry and atomic clocks''. Physical review X 11, 041045 (2021).
https://doi.org/10.1103/PhysRevX.11.041045

[22] Christian D Marciniak, Thomas Feldker, Ivan Pogorelov, Raphael Kaubruegger, Denis V Vasilyev, Rick van Bijnen, Philipp Schindler, Peter Zoller, Rainer Blatt, and Thomas Monz. ``Optimal metrology with programmable quantum sensors''. Nature 603, 604–609 (2022).
https://doi.org/10.1038/s41586-022-04435-4

[23] Vittorio Giovannetti, Seth Lloyd, and Lorenzo Maccone. ``Quantum metrology''. Phys. Rev. Lett. 96, 010401 (2006).
https://doi.org/10.1103/PhysRevLett.96.010401

[24] Rafał Demkowicz-Dobrzański, Jan Kołodyński, and Mădălin Guţă. ``The elusive heisenberg limit in quantum-enhanced metrology''. Nature communications 3, 1063 (2012).
https://doi.org/10.1038/ncomms2067

[25] BM Escher, Ruynet Lima de Matos Filho, and Luiz Davidovich. ``General framework for estimating the ultimate precision limit in noisy quantum-enhanced metrology''. Nature Physics 7, 406–411 (2011).
https://doi.org/10.1038/nphys1958

[26] Rafal Demkowicz-Dobrzański and Lorenzo Maccone. ``Using entanglement against noise in quantum metrology''. Physical review letters 113, 250801 (2014).
https://doi.org/10.1103/PhysRevLett.113.250801

[27] Pavel Sekatski, Michalis Skotiniotis, Janek Kołodyński, and Wolfgang Dür. ``Quantum metrology with full and fast quantum control''. Quantum 1, 27 (2017).
https://doi.org/10.22331/q-2017-09-06-27

[28] Rafał Demkowicz-Dobrzański, Jan Czajkowski, and Pavel Sekatski. ``Adaptive quantum metrology under general markovian noise''. Phys. Rev. X 7, 041009 (2017).
https://doi.org/10.1103/PhysRevX.7.041009

[29] Sisi Zhou, Mengzhen Zhang, John Preskill, and Liang Jiang. ``Achieving the heisenberg limit in quantum metrology using quantum error correction''. Nature communications 9, 78 (2018).
https://doi.org/10.1038/s41467-017-02510-3

[30] S. F. Huelga, C. Macchiavello, T. Pellizzari, A. K. Ekert, M. B. Plenio, and J. I. Cirac. ``Improvement of frequency standards with quantum entanglement''. Phys. Rev. Lett. 79, 3865–3868 (1997).
https://doi.org/10.1103/PhysRevLett.79.3865

[31] Sisi Zhou and Liang Jiang. ``Asymptotic theory of quantum channel estimation''. PRX Quantum 2, 010343 (2021).
https://doi.org/10.1103/PRXQuantum.2.010343

[32] Marcin Jarzyna and Rafał Demkowicz-Dobrzański. ``Matrix product states for quantum metrology''. Phys. Rev. Lett. 110, 240405 (2013).
https://doi.org/10.1103/PhysRevLett.110.240405

[33] Krzysztof Chabuda, Jacek Dziarmaga, Tobias J Osborne, and Rafał Demkowicz-Dobrzański. ``Tensor-network approach for quantum metrology in many-body quantum systems''. Nature communications 11, 250 (2020).
https://doi.org/10.1038/s41467-019-13735-9

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

[35] Carl W Helstrom. ``Quantum detection and estimation theory''. Journal of statistical physics 1, 231–252 (1969).
https://doi.org/10.1007/BF01007479

[36] Alexander S Holevo. ``Probabilistic and statistical aspects of quantum theory''. Volume 1. Springer Science & Business Media. (2011).

[37] Samuel L. Braunstein and Carlton M. Caves. ``Statistical distance and the geometry of quantum states''. Phys. Rev. Lett. 72, 3439–3443 (1994).
https://doi.org/10.1103/PhysRevLett.72.3439

[38] Ole E Barndorff-Nielsen and Richard D Gill. ``Fisher information in quantum statistics''. Journal of Physics A: Mathematical and General 33, 4481 (2000).
https://doi.org/10.1088/0305-4470/33/24/306

[39] Richard D Gill and Serge Massar. ``State estimation for large ensembles''. Physical Review A 61, 042312 (2000).
https://doi.org/10.1103/PhysRevA.61.042312

[40] Matteo G. A. Paris. ``Quantum estimation for quantum technology''. International Journal of Quantum Information 07, 125–137 (2009).
https://doi.org/10.1142/S0219749909004839

[41] Peter W. Shor. ``Scheme for reducing decoherence in quantum computer memory''. Phys. Rev. A 52, R2493–R2496 (1995).
https://doi.org/10.1103/PhysRevA.52.R2493

[42] E. Knill, R. Laflamme, and G. Milburn. ``Efficient linear optics quantum computation'' (2000). quant-ph:quant-ph/0006088.
arXiv:quant-ph/0006088

[43] T. C. Ralph, A. J. F. Hayes, and Alexei Gilchrist. ``Loss-tolerant optical qubits''. Phys. Rev. Lett. 95, 100501 (2005).
https://doi.org/10.1103/PhysRevLett.95.100501

[44] Allen Zang, Tian-Xing Zheng, Peter C. Maurer, Frederic T. Chong, Martin Suchara, and Tian Zhong. ``Enhancing Noisy Quantum Sensing by GHZ State Partitioning'' (2025). arXiv:2507.02829.
arXiv:2507.02829

[45] Masahiro Kitagawa and Masahito Ueda. ``Squeezed spin states''. Phys. Rev. A 47, 5138–5143 (1993).
https://doi.org/10.1103/PhysRevA.47.5138

[46] Jian Ma, Xiaoguang Wang, C.P. Sun, and Franco Nori. ``Quantum spin squeezing''. Physics Reports 509, 89–165 (2011).
https://doi.org/10.1016/j.physrep.2011.08.003

[47] Duger Ulam-Orgikh and Masahiro Kitagawa. ``Spin squeezing and decoherence limit in ramsey spectroscopy''. Physical Review A 64, 052106 (2001).
https://doi.org/10.1103/PhysRevA.64.052106

[48] Jonatan Bohr Brask, Rafael Chaves, and Janek Kołodyński. ``Improved quantum magnetometry beyond the standard quantum limit''. Physical Review X 5, 031010 (2015).
https://doi.org/10.1103/PhysRevX.5.031010

[49] Anders S Sørensen and Klaus Mølmer. ``Entanglement and extreme spin squeezing''. Physical review letters 86, 4431 (2001).
https://doi.org/10.1103/PhysRevLett.86.4431

[50] Géza Tóth and Iagoba Apellaniz. ``Quantum metrology from a quantum information science perspective''. Journal of Physics A: Mathematical and Theoretical 47, 424006 (2014).
https://doi.org/10.1088/1751-8113/47/42/424006

[51] Luca Pezzé and Augusto Smerzi. ``Entanglement, nonlinear dynamics, and the heisenberg limit''. Physical review letters 102, 100401 (2009).
https://doi.org/10.1103/PhysRevLett.102.100401

[52] Elliott H. Lieb and Derek W. Robinson. ``The finite group velocity of quantum spin systems''. Commun. Math. Phys. 28, 251–257 (1972).
https://doi.org/10.1007/BF01645779

[53] Chi-Fang (Anthony) Chen, Andrew Lucas, and Chao Yin. ``Speed limits and locality in many-body quantum dynamics''. Reports on Progress in Physics 86, 116001 (2023).
https://doi.org/10.1088/1361-6633/acfaae

[54] S. Bravyi, M. B. Hastings, and F. Verstraete. ``Lieb-robinson bounds and the generation of correlations and topological quantum order''. Phys. Rev. Lett. 97, 050401 (2006).
https://doi.org/10.1103/PhysRevLett.97.050401

[55] Yaoming Chu, Xiangbei Li, and Jianming Cai. ``Strong quantum metrological limit from many-body physics''. Phys. Rev. Lett. 130, 170801 (2023).
https://doi.org/10.1103/PhysRevLett.130.170801

[56] B L Higgins, D W Berry, S D Bartlett, M W Mitchell, H M Wiseman, and G J Pryde. ``Demonstrating heisenberg-limited unambiguous phase estimation without adaptive measurements''. New Journal of Physics 11, 073023 (2009).
https://doi.org/10.1088/1367-2630/11/7/073023

[57] Shelby Kimmel, Guang Hao Low, and Theodore J. Yoder. ``Robust calibration of a universal single-qubit gate set via robust phase estimation''. Phys. Rev. A 92, 062315 (2015).
https://doi.org/10.1103/PhysRevA.92.062315

[58] Federico Belliardo and Vittorio Giovannetti. ``Achieving heisenberg scaling with maximally entangled states: An analytic upper bound for the attainable root-mean-square error''. Phys. Rev. A 102, 042613 (2020).
https://doi.org/10.1103/PhysRevA.102.042613

[59] Tommaso Macrì, Augusto Smerzi, and Luca Pezzè. ``Loschmidt echo for quantum metrology''. Phys. Rev. A 94, 010102 (2016).
https://doi.org/10.1103/PhysRevA.94.010102

[60] David Poulin. ``Lieb-robinson bound and locality for general markovian quantum dynamics''. Phys. Rev. Lett. 104, 190401 (2010).
https://doi.org/10.1103/PhysRevLett.104.190401

[61] Jae-Seung Lee and A. K. Khitrin. ``Stimulated wave of polarization in a one-dimensional ising chain''. Phys. Rev. A 71, 062338 (2005).
https://doi.org/10.1103/PhysRevA.71.062338

[62] Atsuki Yoshinaga, Mamiko Tatsuta, and Yuichiro Matsuzaki. ``Entanglement-enhanced sensing using a chain of qubits with always-on nearest-neighbor interactions''. Phys. Rev. A 103, 062602 (2021).
https://doi.org/10.1103/PhysRevA.103.062602

[63] Arkadiusz Kobus and Rafał Demkowicz-Dobrzański. ``Asymptotically optimal joint phase and dephasing strength estimation using spin-squeezed states'' (2025).
arXiv:2507.22997

[64] Maxwell Block, Bingtian Ye, Brenden Roberts, Sabrina Chern, Weijie Wu, Zilin Wang, Lode Pollet, Emily J Davis, Bertrand I Halperin, and Norman Y Yao. ``Scalable spin squeezing from finite-temperature easy-plane magnetism''. Nature PhysicsPages 1–7 (2024).
https://doi.org/10.1038/s41567-024-02562-5

[65] Tommaso Comparin, Fabio Mezzacapo, and Tommaso Roscilde. ``Robust spin squeezing from the tower of states of u(1)-symmetric spin hamiltonians''. Phys. Rev. A 105, 022625 (2022).
https://doi.org/10.1103/PhysRevA.105.022625

[66] Tommaso Roscilde, Filippo Caleca, Adriano Angelone, and Fabio Mezzacapo. ``Scalable spin squeezing from critical slowing down in short-range interacting systems''. Phys. Rev. Lett. 133, 210401 (2024).
https://doi.org/10.1103/PhysRevLett.133.210401

[67] R. Augusiak, J. Kołodyński, A. Streltsov, M. N. Bera, A. Acín, and M. Lewenstein. ``Asymptotic role of entanglement in quantum metrology''. Phys. Rev. A 94, 012339 (2016).
https://doi.org/10.1103/PhysRevA.94.012339

[68] A. T. Rezakhani, M. Hassani, and S. Alipour. ``Continuity of the quantum fisher information''. Phys. Rev. A 100, 032317 (2019).
https://doi.org/10.1103/PhysRevA.100.032317

[69] Rajendra Bhatia. ``Pinching, trimming, truncating, and averaging of matrices''. The American Mathematical Monthly 107, 602–608 (2000).
https://doi.org/10.1080/00029890.2000.12005245

[70] E. P. Wigner and Mutsuo M. Yanase. ``Information contents of distributions''. Proceedings of the National Academy of Sciences 49, 910–918 (1963).
https://doi.org/10.1073/pnas.49.6.910

[71] Shunlong Luo. ``Wigner-yanase skew information vs. quantum fisher information''. Proceedings of the American Mathematical Society 132, 885–890 (2004).
https://doi.org/10.1090/S0002-9939-03-07175-2

[72] Benoit Roubert, Petr Braun, and Daniel Braun. ``Large effects of boundaries on spin amplification in spin chains''. Phys. Rev. A 82, 022302 (2010).
https://doi.org/10.1103/PhysRevA.82.022302

[73] Kianna Wan and Robert Lasenby. ``Bounds on adaptive quantum metrology under markovian noise''. Physical Review Research 4, 033092 (2022).
https://doi.org/10.1103/PhysRevResearch.4.033092

Cited by

[1] Cheng-Ju Lin, Zi-Wen Liu, Victor V. Albert, and Alexey V. Gorshkov, "Covariant Quantum Error-Correcting Codes with Metrological Entanglement Advantage", Physical Review Letters 135 11, 110801 (2025).

[2] Ben T. McDonough, Chao Yin, Andrew Lucas, and Carolyn Zhang, "Lieb-Robinson Bounds with Exponential-in-Volume Tails", PRX Quantum 6 4, 040322 (2025).

[3] Anthony J. Brady, Yu-Xin Wang, Victor V. Albert, Alexey V. Gorshkov, and Quntao Zhuang, "Correlated Noise Estimation with Quantum Sensor Networks", Physical Review Letters 136 8, 080803 (2026).

[4] Allen Zang, Tian-Xing Zheng, Peter C. Maurer, Frederic T. Chong, Martin Suchara, and Tian Zhong, "Enhancing Noisy Quantum Sensing by GHZ State Partitioning", arXiv:2507.02829, (2025).

[5] Yu-Xin Wang, Flavio Salvati, David R. M. Arvidsson-Shukur, William F. Braasch, Kater Murch, and Nicole Yunger Halpern, "Quantum metrology enhanced by effective time reversal", arXiv:2601.20952, (2026).

The above citations are from SAO/NASA ADS (last updated successfully 2026-08-08 19:10:12). The list may be incomplete as not all publishers provide suitable and complete citation data.

On Crossref's cited-by service no data on citing works was found (last attempt 2026-08-08 19:10:11).

This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.