- Physical Review Journals
All JournalsPhysics Magazine

PRX Quantum

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Randomized Measurements for Multiparameter Quantum Metrology

Sisi Zhou1,2,* and Senrui Chen3,4,†

  • *Contact author: sisi.zhou26@gmail.com
  • †Contact author: csenrui@gmail.com
PDF

PRX Quantum 7, 010314 – Published 21 January, 2026

DOI: https://doi.org/10.1103/s27y-gbrp

Abstract

The optimal quantum measurements for estimating different unknown parameters in a parameterized quantum state are usually incompatible with each other. Traditional approaches to addressing the measurement incompatibility issue, such as the Holevo Cramér-Rao bound, suffer from multiple difficulties toward practical applicability, as the optimal measurement strategies are usually state-dependent, difficult to implement and also take complex analyses to determine. Here we study randomized measurements as an approach for multiparameter quantum metrology. We show quantum measurements on single copies of quantum states given by 3-designs perform near-optimally when estimating an arbitrary number of parameters in pure states and more generally, approximately low-rank well-conditioned states, whose metrological information is largely concentrated in a low-dimensional subspace. The near-optimality is also shown in estimating the maximal number of parameters for three types of mixed states that are well conditioned on their supports. Examples of fidelity estimation and Hamiltonian estimation are explicitly provided to demonstrate the power and limitation of randomized measurements in multiparameter quantum metrology.

View figure in article

Physics Subject Headings (PhySH)

Popular Summary

Measuring multiple unknown parameters in a quantum system—known as multiparameter quantum metrology—is notoriously difficult. The main challenge is that the best measurements for different parameters are often incompatible with each other, making it impossible to implement them all at once. Traditional solutions exist, but they typically require complicated, state-dependent procedures or collective measurements on many copies of a quantum state-tasks that pose significant experimental challenges.

In this work, we show that a simple idea can overcome these obstacles: just use randomized measurements. Instead of carefully designing the “perfect” measurement for each problem, we choose measurements at random from a well-defined statistical pattern known as a 3-design, which can be efficiently realized using random Clifford circuits on quantum devices.

Our main theoretical result is that these randomized measurements perform nearly as well as the optimal possible measurements-within a constant factor-for a wide range of important quantum states, including all pure states and many low-rank mixed states. In other words, randomness can achieve almost the same precision as highly optimized strategies, while being dramatically easier to implement. We also identify clear limits: for certain mixed states, randomized measurements lose their advantage.

Beyond technical results, our findings reveal a new and counterintuitive principle in quantum metrology-sometimes, randomness is almost as powerful as perfection. This insight connects ideas from quantum learning-where randomness is a key resource-with the goals of precision quantum sensing, opening a practical route toward scalable, experiment-friendly quantum metrology.

Article Text

References (116)

  1. V. Giovannetti, S. Lloyd, and L. Maccone, Advances in quantum metrology, Nat. Photonics 5, 222 (2011).
  2. C. L. Degen, F. Reinhard, and P. Cappellaro, Quantum sensing, Rev. Mod. Phys. 89, 035002 (2017).
  3. 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).
  4. S. Pirandola, B. R. Bardhan, T. Gehring, C. Weedbrook, and S. Lloyd, Advances in photonic quantum sensing, Nat. Photonics 12, 724 (2018).
  5. C. M. Caves, Quantum-mechanical noise in an interferometer, Phys. Rev. D 23, 1693 (1981).
  6. B. Yurke, S. L. McCall, and J. R. Klauder, Su(2) and SU(1,1) interferometers, Phys. Rev. A 33, 4033 (1986).
  7. LIGO Collaboration, A gravitational wave observatory operating beyond the quantum shot-noise limit, Nat. Phys. 7, 962 (2011).
  8. LIGO Collaboration, Enhanced sensitivity of the LIGO gravitational wave detector by using squeezed states of light, Nat. Photonics 7, 613 (2013).
  9. 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 (1992).
  10. 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).
  11. 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 (2004).
  12. J. Taylor, P. Cappellaro, L. Childress, L. Jiang, D. Budker, P. Hemmer, A. Yacoby, R. Walsworth, and M. Lukin, High-sensitivity diamond magnetometer with nanoscale resolution, Nat. Phys. 4, 810 (2008).
  13. H. Zhou, J. Choi, S. Choi, R. Landig, A. M. Douglas, J. Isoya, F. Jelezko, S. Onoda, H. Sumiya, P. Cappellaro et al., Quantum metrology with strongly interacting spin systems, Phys. Rev. X 10, 031003 (2020).
  14. T. Rosenband, D. Hume, P. Schmidt, C.-W. Chou, A. Brusch, L. Lorini, W. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker et al., Frequency ratio of Al+ and Hg+ single-ion optical clocks; metrology at the 17th decimal place, Science 319, 1808 (2008).
  15. J. Appel, P. J. Windpassinger, D. Oblak, U. B. Hoff, N. Kjærgaard, and E. S. Polzik, Mesoscopic atomic entanglement for precision measurements beyond the standard quantum limit, Proc. Natl. Acad. Sci. U.S.A. 106, 10960 (2009).
  16. A. D. Ludlow, M. M. Boyd, J. Ye, E. Peik, and P. O. Schmidt, Optical atomic clocks, Rev. Mod. Phys. 87, 637 (2015).
  17. R. Kaubruegger, D. V. Vasilyev, M. Schulte, K. Hammerer, and P. Zoller, Quantum variational optimization of Ramsey interferometry and atomic clocks, Phys. Rev. X 11, 041045 (2021).
  18. C. D. Marciniak, T. Feldker, I. Pogorelov, R. Kaubruegger, D. V. Vasilyev, R. van Bijnen, P. Schindler, P. Zoller, R. Blatt, and T. Monz, Optimal metrology with programmable quantum sensors, Nature 603, 604 (2022).
  19. D. Le Sage, K. Arai, D. R. Glenn, S. J. DeVience, L. M. Pham, L. Rahn-Lee, M. D. Lukin, A. Yacoby, A. Komeili, and R. L. Walsworth, Optical magnetic imaging of living cells, Nature 496, 486 (2013).
  20. G. B. Lemos, V. Borish, G. D. Cole, S. Ramelow, R. Lapkiewicz, and A. Zeilinger, Quantum imaging with undetected photons, Nature 512, 409 (2014).
  21. M. Tsang, R. Nair, and X.-M. Lu, Quantum theory of superresolution for two incoherent optical point sources, Phys. Rev. X 6, 031033 (2016).
  22. M. Abobeih, J. Randall, C. Bradley, H. Bartling, M. Bakker, M. Degen, M. Markham, D. Twitchen, and T. Taminiau, Atomic-scale imaging of a 27-nuclear-spin cluster using a quantum sensor, Nature 576, 411 (2019).
  23. C. W. Helstrom, Minimum mean-squared error of estimates in quantum statistics, Phys. Lett. A 25, 101 (1967).
  24. C. W. Helstrom, The minimum variance of estimates in quantum signal detection, IEEE Trans. Inf. Theory 14, 234 (1968).
  25. C. W. Helstrom, Quantum Detection and Estimation Theory (Academic Press, New York City, USA, 1976).
  26. A. S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory (Springer Science & Business Media, Berlin, Germany, 2011), Vol. 1.
  27. S. L. Braunstein and C. M. Caves, Statistical distance and the geometry of quantum states, Phys. Rev. Lett. 72, 3439 (1994).
  28. O. E. Barndorff-Nielsen and R. D. Gill, Fisher information in quantum statistics, J. Phys. A: Math. Gen. 33, 4481 (2000).
  29. M. G. Paris, Quantum estimation for quantum technology, Int. J. Quantum Inf. 7, 125 (2009).
  30. R. D. Gill and S. Massar, State estimation for large ensembles, Phys. Rev. A 61, 042312 (2000).
  31. K. Matsumoto, A new approach to the Cramér-Rao-type bound of the pure-state model, J. Phys. A: Math. Gen. 35, 3111 (2002).
  32. H. Nagaoka, in Asymptotic Theory of Quantum Statistical Inference: Selected Papers (World Scientific, Hackensack, New Jersey, USA, 2005), p. 100.
  33. L. O. Conlon, J. Suzuki, P. K. Lam, and S. M. Assad, Efficient computation of the Nagaoka–Hayashi bound for multiparameter estimation with separable measurements, Npj Quantum Inf. 7, 110 (2021).
  34. S. Ragy, M. Jarzyna, and R. Demkowicz-Dobrzański, Compatibility in multiparameter quantum metrology, Phys. Rev. A 94, 052108 (2016).
  35. R. Demkowicz-Dobrzański, W. Górecki, and M. Guţă, Multi-parameter estimation beyond quantum fisher information, J. Phys. A: Math. Theor. 53, 363001 (2020).
  36. M. Szczykulska, T. Baumgratz, and A. Datta, Multi-parameter quantum metrology, Adv. Phys.: X 1, 621 (2016).
  37. J. Liu, H. Yuan, X.-M. Lu, and X. Wang, Quantum fisher information matrix and multiparameter estimation, J. Phys. A: Math. Theor. 53, 023001 (2020).
  38. J. S. Sidhu and P. Kok, Geometric perspective on quantum parameter estimation, AVS Quantum Sci. 2, 014701 (2020).
  39. J. Kahn and M. Guţă, Local asymptotic normality for finite dimensional quantum systems, Commun. Math. Phys. 289, 597 (2009).
  40. K. Yamagata, A. Fujiwara, and R. D. Gill, Quantum local asymptotic normality based on a new quantum likelihood ratio, Ann. Stat. 41, 2197 (2013).
  41. Y. Yang, G. Chiribella, and M. Hayashi, Attaining the ultimate precision limit in quantum state estimation, Commun. Math. Phys. 368, 223 (2019).
  42. P. C. Humphreys, M. Barbieri, A. Datta, and I. A. Walmsley, Quantum enhanced multiple phase estimation, Phys. Rev. Lett. 111, 070403 (2013).
  43. M. D. Vidrighin, G. Donati, M. G. Genoni, X.-M. Jin, W. S. Kolthammer, M. Kim, A. Datta, M. Barbieri, and I. A. Walmsley, Joint estimation of phase and phase diffusion for quantum metrology, Nat. Commun. 5, 3532 (2014).
  44. J. Suzuki, Explicit formula for the Holevo bound for two-parameter qubit-state estimation problem, J. Math. Phys. 57, 042201 (2016).
  45. Y. Chen and H. Yuan, Maximal quantum fisher information matrix, New J. Phys. 19, 063023 (2017).
  46. L. Pezzè, M. A. Ciampini, N. Spagnolo, P. C. Humphreys, A. Datta, I. A. Walmsley, M. Barbieri, F. Sciarrino, and A. Smerzi, Optimal measurements for simultaneous quantum estimation of multiple phases, Phys. Rev. Lett. 119, 130504 (2017).
  47. N. Li, C. Ferrie, J. A. Gross, A. Kalev, and C. M. Caves, Fisher-symmetric informationally complete measurements for pure states, Phys. Rev. Lett. 116, 180402 (2016).
  48. H. Zhu and M. Hayashi, Universally fisher-symmetric informationally complete measurements, Phys. Rev. Lett. 120, 030404 (2018).
  49. C. Vargas, L. Pereira, and A. Delgado, Near-optimal pure state estimation with adaptive fisher-symmetric measurements, arXiv:2412.04555.
  50. J. Yang, S. Pang, Y. Zhou, and A. N. Jordan, Optimal measurements for quantum multiparameter estimation with general states, Phys. Rev. A 100, 032104 (2019).
  51. A. Carollo, B. Spagnolo, A. A. Dubkov, and D. Valenti, On quantumness in multi-parameter quantum estimation, J. Stat. Mech.: Theory Exp. 2019, 094010 (2019).
  52. F. Albarelli, M. Barbieri, M. G. Genoni, and I. Gianani, A perspective on multiparameter quantum metrology: From theoretical tools to applications in quantum imaging, Phys. Lett. A 384, 126311 (2020).
  53. J. Suzuki, Y. Yang, and M. Hayashi, Quantum state estimation with nuisance parameters, J. Phys. A: Math. Theor. 53, 453001 (2020).
  54. Z. Hou, Z. Zhang, G.-Y. Xiang, C.-F. Li, G.-C. Guo, H. Chen, L. Liu, and H. Yuan, Minimal tradeoff and ultimate precision limit of multiparameter quantum magnetometry under the parallel scheme, Phys. Rev. Lett. 125, 020501 (2020).
  55. F. Belliardo and V. Giovannetti, Incompatibility in quantum parameter estimation, New J. Phys. 23, 063055 (2021).
  56. A. Candeloro, M. G. Paris, and M. G. Genoni, On the properties of the asymptotic incompatibility measure in multiparameter quantum estimation, J. Phys. A: Math. Theor. 54, 485301 (2021).
  57. X.-M. Lu and X. Wang, Incorporating Heisenberg’s uncertainty principle into quantum multiparameter estimation, Phys. Rev. Lett. 126, 120503 (2021).
  58. H. Chen, Y. Chen, and H. Yuan, Information geometry under hierarchical quantum measurement, Phys. Rev. Lett. 128, 250502 (2022).
  59. H. Chen, Y. Chen, and H. Yuan, Incompatibility measures in multiparameter quantum estimation under hierarchical quantum measurements, Phys. Rev. A 105, 062442 (2022).
  60. S. K. Yung, L. O. Conlon, J. Zhao, P. K. Lam, and S. M. Assad, Comparison of estimation limits for quantum two-parameter estimation, Phys. Rev. Res. 6, 033315 (2024).
  61. F. Albarelli, J. F. Friel, and A. Datta, Evaluating the Holevo Cramér-Rao bound for multiparameter quantum metrology, Phys. Rev. Lett. 123, 200503 (2019).
  62. W. Górecki, S. Zhou, L. Jiang, and R. Demkowicz-Dobrzański, Optimal probes and error-correction schemes in multi-parameter quantum metrology, Quantum 4, 288 (2020).
  63. M. Tsang, F. Albarelli, and A. Datta, Quantum semiparametric estimation, Phys. Rev. X 10, 031023 (2020).
  64. J. S. Sidhu, Y. Ouyang, E. T. Campbell, and P. Kok, Tight bounds on the simultaneous estimation of incompatible parameters, Phys. Rev. X 11, 011028 (2021).
  65. M. Hayashi and Y. Ouyang, Tight Cramér-Rao type bounds for multiparameter quantum metrology through conic programming, Quantum 7, 1094 (2023).
  66. J. W. Gardner, T. Gefen, S. A. Haine, J. J. Hope, and Y. Chen, Achieving the fundamental quantum limit of linear waveform estimation, Phys. Rev. Lett. 132, 130801 (2024).
  67. A. Elben, S. T. Flammia, H.-Y. Huang, R. Kueng, J. Preskill, B. Vermersch, and P. Zoller, The randomized measurement toolbox, Nat. Rev. Phys. 5, 9 (2023).
  68. J. Watrous, The Theory of Quantum Information (Cambridge University Press, Cambridge, United Kingdom, 2018).
  69. E. Knill, D. Leibfried, R. Reichle, J. Britton, R. B. Blakestad, J. D. Jost, C. Langer, R. Ozeri, S. Seidelin, and D. J. Wineland, Randomized benchmarking of quantum gates, Phys. Rev. A 77, 012307 (2008).
  70. C. Dankert, R. Cleve, J. Emerson, and E. Livine, Exact and approximate unitary 2-designs and their application to fidelity estimation, Phys. Rev. A 80, 012304 (2009).
  71. J. Emerson, R. Alicki, and K. Życzkowski, Scalable noise estimation with random unitary operators, J. Opt. B: Quantum Semiclass. Opt. 7, S347 (2005).
  72. H.-Y. Huang, R. Kueng, and J. Preskill, Predicting many properties of a quantum system from very few measurements, Nat. Phys. 16, 1050 (2020).
  73. T. Brydges, A. Elben, P. Jurcevic, B. Vermersch, C. Maier, B. P. Lanyon, P. Zoller, R. Blatt, and C. F. Roos, Probing Rényi entanglement entropy via randomized measurements, Science 364, 260 (2019).
  74. J. M. Renes, R. Blume-Kohout, A. J. Scott, and C. M. Caves, Symmetric informationally complete quantum measurements, J. Math. Phys. 45, 2171 (2004).
  75. A. Ambainis and J. Emerson, in Twenty-Second Annual IEEE Conference on Computational Complexity (CCC’07) (IEEE, San Diego, California, USA, 2007), p. 129.
  76. A. Roy and A. J. Scott, Unitary designs and codes, Des. Codes Cryptogr. 53, 13 (2009).
  77. D. Gross, K. Audenaert, and J. Eisert, Evenly distributed unitaries: On the structure of unitary designs, J. Math. Phys. 48, 052104 (2007).
  78. R. Kueng and D. Gross, Qubit stabilizer states are complex projective 3-designs, arXiv:1510.02767.
  79. Z. Webb, The Clifford group forms a unitary 3-design, Quantum Inf. Comput. 16, 1379 (2016).
  80. H. Zhu, Multiqubit clifford groups are unitary 3-designs, Phys. Rev. A 96, 062336 (2017).
  81. S. Aaronson and D. Gottesman, Improved simulation of stabilizer circuits, Phys. Rev. A 70, 052328 (2004).
  82. S. Bravyi and D. Maslov, Hadamard-free circuits expose the structure of the clifford group, IEEE Trans. Inf. Theory 67, 4546 (2021).
  83. C. Moore and M. Nilsson, Parallel quantum computation and quantum codes, SIAM J. Comput. 31, 799 (2001).
  84. J. Jiang, X. Sun, S.-H. Teng, B. Wu, K. Wu, and J. Zhang, in Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms (SIAM, Seattle, Washington, USA, 2020), p. 213.
  85. F. G. Brandao, A. W. Harrow, and M. Horodecki, Local random quantum circuits are approximate polynomial-designs, Commun. Math. Phys. 346, 397 (2016).
  86. J. Haferkamp, F. Montealegre-Mora, M. Heinrich, J. Eisert, D. Gross, and I. Roth, Efficient unitary designs with a system-size independent number of non-clifford gates, Commun. Math. Phys. 397, 995 (2023).
  87. A. W. Harrow and S. Mehraban, Approximate unitary t-designs by short random quantum circuits using nearest-neighbor and long-range gates, Commun. Math. Phys. 401, 1531 (2023).
  88. T. Schuster, J. Haferkamp, and H.-Y. Huang, Random unitaries in extremely low depth, Science 389, 92 (2025).
  89. B. Bajnok, Constructions of spherical 3-designs, Graphs Combin. 14, 97 (1998).
  90. E. Bannai, M. Nakahara, D. Zhao, and Y. Zhu, On the explicit constructions of certain unitary t-designs, J. Phys. A: Math. Theor. 52, 495301 (2019).
  91. E. Bannai, Y. Nakata, T. Okuda, and D. Zhao, Explicit construction of exact unitary designs, Adv. Math. 405, 108457 (2022).
  92. C. R. Rao, C. R. Rao, M. Statistiker, C. R. Rao, and C. R. Rao, Linear Statistical Inference and Its Applications (Wiley New York, New York City, USA, 1973), Vol. 2.
  93. S. M. Kay, Fundamentals of Statistical Signal Processing: Volume I Estimation Theory (Prentice-Hall, Inc., Englewood Cliffs, New Jersey, USA, 1993).
  94. E. L. Lehmann and G. Casella, Theory of Point Estimation (Springer Science & Business Media, Berlin, Germany, 2006).
  95. D. R. Cox, Inference and Asymptotics (Routledge, London, United Kingdom, 2017).
  96. G. Casella and R. L. Berger, Statistical Inference (Duxbury, Pacific Grove, CA, 2002), Vol. 2.
  97. S. Chen, J. Cotler, H.-Y. Huang, and J. Li, in 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS) (IEEE, Denver, Colorado, USA, 2022), p. 574.
  98. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, United Kingdom, 2010).
  99. H. Nagaoka, in Asymptotic Theory of Quantum Statistical Inference: Selected Papers (World Scientific, Hackensack, New Jersey, USA, 2005), p. 125.
  100. Z. Hou, J.-F. Tang, J. Shang, H. Zhu, J. Li, Y. Yuan, K.-D. Wu, G.-Y. Xiang, C.-F. Li, and G.-C. Guo, Deterministic realization of collective measurements via photonic quantum walks, Nat. Commun. 9, 1414 (2018).
  101. M. Hayashi, Asymptotic estimation theory for a finite-dimensional pure state model, J. Phys. A: Math. Gen. 31, 4633 (1998).
  102. H. Zhu, Quantum state estimation and symmetric informationally complete poms (2012), available online at http://scholarbank.nus.edu.sg/bitstream/10635/35247/1/ZhuHJthesis.pdf.
  103. H. Zhu, Quantum state estimation with informationally overcomplete measurements, Phys. Rev. A 90, 012115 (2014).
  104. J. Lu and K. Sha, Quantum fisher information matrix via its classical counterpart from random measurements, arXiv:2509.08196.
  105. D. A. Roberts and B. Yoshida, Chaos and complexity by design, J. High Energy Phys. 2017, 1 (2017).
  106. B. Efron and R. J. Tibshirani, An Introduction to the Bootstrap (Chapman and Hall/CRC, London, United Kingdom, 1994).
  107. J. L. Hintze and R. D. Nelson, Violin plots: A box plot-density trace synergism, Am. Stat. 52, 181 (1998).
  108. A. Javadi-Abhari, M. Treinish, K. Krsulich, C. J. Wood, J. Lishman, J. Gacon, S. Martiel, P. D. Nation, L. S. Bishop, A. W. Cross et al., Quantum computing with Qiskit, arXiv:2405.08810.
  109. All functions we calculate in this section will be implicitly evaluated at θ=0, unless specified otherwise.
  110. This notation should be distinguished from J[⋆] which is the QFIM of the parameterized state ⋆ that we use later.
  111. Here it is not suitable to choose the cost matrix W=J(ψ) as in the definition of weak near-optimality, because we aim to compare different input states (for channel estimation) instead of different POVMs (for state estimation) and the cost matrix should not depend on the input state. W=1 is the most natural choice as it assigns equal importance to all parameters.
  112. W. Górecki, R. Demkowicz-Dobrzański, H. M. Wiseman, and D. W. Berry, π-corrected Heisenberg limit, Phys. Rev. Lett. 124, 030501 (2020).
  113. J. J. Meyer, S. Khatri, D. Stilck França, J. Eisert, and P. Faist, Quantum metrology in the finite-sample regime, PRX Quantum 6, 030336 (2025).
  114. https://github.com/csenrui/random-measurement-metrology.
  115. Note that in contrast to the Fisher-symmetry condition which defines the optimality of measurement in an exact fashion, the weak near-optimality is defined in an approximate fashion (i.e., we tolerate any constant-factor suboptimality). Therefore, we cannot hope for an exact equivalence between the Fisher-symmetry condition and the weak near-optimality condition. If we fix c1=c2=1 in the definition of weak near-optimality, the equivalence will be exact.
  116. E. W. Weisstein, Chi-squared distribution, https://mathworld.wolfram.com (2002).

Outline

Information

PRX Quantum 7, 010314– Published 21 January, 2026
  • Received 16 February 2025
  • Revised 6 November 2025
  • Accepted 15 December 2025
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation