Unlocking Heisenberg Sensitivity with Sequential Weak Measurement Preparation

T. B. Lantaño; Dayou Yang; K. M. R. Audenaert; S. F. Huelga; M. B. Plenio

doi:10.22331/q-2025-01-14-1590

Unlocking Heisenberg Sensitivity with Sequential Weak Measurement Preparation

T. B. Lantaño, Dayou Yang, K. M. R. Audenaert, S. F. Huelga, and M. B. Plenio

Institut für Theoretische Physik, Albert-Einstein-Allee 11, Universität Ulm 89069, Germany

Published:	2025-01-14, volume 9, page 1590
Editor:	Pérola Milman
Eprint:	arXiv:2403.05954v3
Doi:	https://doi.org/10.22331/q-2025-01-14-1590
Citation:	Quantum 9, 1590 (2025).

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

Abstract

We propose a state preparation protocol based on sequential measurements of a central spin coupled with a spin ensemble, and investigate the usefulness of the generated multi-spin states for quantum enhanced metrology. Our protocol is shown to generate highly entangled spin states, devoid of the necessity for non-linear spin interactions. The metrological sensitivity of the resulting state surpasses the standard quantum limit, reaching the Heisenberg limit under symmetric coupling strength conditions. We also explore asymmetric coupling strengths, identifying specific preparation windows in time for optimal sensitivity. Our findings introduce a novel method for generating large-scale, non-classical, entangled states, enabling quantum-enhanced metrology within current experimental capabilities.

► BibTeX data

@article{Lantano2025unlockingheisenberg,
  doi = {10.22331/q-2025-01-14-1590},
  url = {https://doi.org/10.22331/q-2025-01-14-1590},
  title = {Unlocking {H}eisenberg {S}ensitivity with {S}equential {W}eak {M}easurement {P}reparation},
  author = {Lanta{\~{n}}o, T. B. and Yang, Dayou and Audenaert, K. M. R. and Huelga, S. F. and Plenio, M. B.},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1590},
  month = jan,
  year = {2025}
}

► References

[1] C. M. Caves, Quantum-mechanical noise in an interferometer, Phys. Rev. D 23, 1693 (1981).
https://doi.org/10.1103/PhysRevD.23.1693

[2] M. Kitagawa and M. Ueda, Squeezed spin states, Phys. Rev. A 47, 5138 (1993).
https://doi.org/10.1103/PhysRevA.47.5138

[3] D. Heinzen, Squeezed atomic states and projection noise in spectroscopy, Physical Review A 50, 67 (1994).
https://doi.org/10.1103/PhysRevA.50.67

[4] 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 (1997).
https://doi.org/10.1103/PhysRevLett.79.3865

[5] P. Zanardi, M. G. Paris, and L. C. Venuti, Quantum criticality as a resource for quantum estimation, Physical Review A 78, 042105 (2008).
https://doi.org/10.1103/PhysRevA.78.042105

[6] M. Tsang, Quantum transition-edge detectors, Physical Review A 88, 021801 (2013).
https://doi.org/10.1103/PhysRevA.88.021801

[7] Y. Chu, S. Zhang, B. Yu, and J. Cai, Dynamic framework for criticality-enhanced quantum sensing, Physical Review Letters 126, 010502 (2021).
https://doi.org/10.1103/PhysRevLett.126.010502

[8] T. Ilias, D. Yang, S. F. Huelga, and M. B. Plenio, Criticality-enhanced quantum sensing via continuous measurement, PRX Quantum 3, 010354 (2022).
https://doi.org/10.1103/PRXQuantum.3.010354

[9] D. Yang, S. F. Huelga, and M. B. Plenio, Efficient information retrieval for sensing via continuous measurement, Physical Review X 13, 031012 (2023).
https://doi.org/10.1103/PhysRevX.13.031012

[10] J. F. Haase, A. Smirne, J. Kołodynski, R. Demkowicz-Dobrzanski, and S. F. Huelga, Precision limits in quantum metrology with open quantum systems, Quantum Measurements and Quantum Metrology 5, 13 (2016).
https://doi.org/10.1515/qmetro-2018-0002

[11] V. Giovannetti, S. Lloyd, and L. Maccone, Quantum metrology, Phys. Rev. Lett. 96, 010401 (2006).
https://doi.org/10.1103/PhysRevLett.96.010401

[12] 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).
https://doi.org/10.1103/PhysRevA.46.R6797

[13] A. Sørensen, L.-M. Duan, J. I. Cirac, and P. Zoller, Many-particle entanglement with bose–einstein condensates, Nature 409, 63–66 (2001).
https://doi.org/10.1038/35051038

[14] 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 464, 1170 (2010).
https://doi.org/10.1038/nature08988

[15] K. Helmerson and L. You, Creating massive entanglement of bose-einstein condensed atoms, Phys. Rev. Lett. 87, 170402 (2001).
https://doi.org/10.1103/PhysRevLett.87.170402

[16] C. Gross, Spin squeezing, entanglement and quantum metrology with bose–einstein condensates, J. Phys. B: At. Mol. Opt. Phys. 45, 103001 (2012).
https://doi.org/10.1088/0953-4075/45/10/103001

[17] G. S. Agarwal, R. R. Puri, and R. P. Singh, Atomic schrödinger cat states, Phys. Rev. A 56, 2249 (1997).
https://doi.org/10.1103/PhysRevA.56.2249

[18] B. C. Sanders and C. C. Gerry, Connection between the noon state and a superposition of su(2) coherent states, Phys. Rev. A 90, 045804 (2014).
https://doi.org/10.1103/PhysRevA.90.045804

[19] S. Sharma, V. A. S. V. Bittencourt, A. D. Karenowska, , and S. V. Kusminskiy, Spin cat states in ferromagnetic insulators, Phys. Rev. B 103, L100403 (2021).
https://doi.org/10.1103/PhysRevB.103.L100403

[20] J. Huang, M. Zhuang, B. Lu, Y. Ke, and C. Lee, Achieving heisenberg-limited metrology with spin cat states via interaction-based readout, Phys. Rev. A 98, 012129 (2018).
https://doi.org/10.1103/PhysRevA.98.012129

[21] J. Huang, X. Qin, H. Zhong, Y. Ke, and C. Lee, Quantum metrology with spin cat states under dissipation, Scientific Reports 5, 17894 (2015).
https://doi.org/10.1038/srep17894

[22] C. Lee, Adiabatic mach-zehnder interferometry on a quantized bose-josephson junction, Phys. Rev. Lett. 97, 150402 (2006).
https://doi.org/10.1103/PhysRevLett.97.150402

[23] X.-Y. Luo, Y.-Q. Zou, L.-N. Wu, Q. Liu, M.-F. Han, M. K. Tey, and L. You, Deterministic entanglement generation from driving through quantum phase transitions, Science 355, 620 (2017).
https://doi.org/10.1126/science.aag1106

[24] M. Zhuang, J. Huang, Y. Ke, and C. Lee, Quantum adiabatic evolution: Symmetry-protected quantum adiabatic evolution in spontaneous symmetry-breaking transitions, Ann. Phys. (Berlin) 532, 1900471 (2020).
https://doi.org/10.1002/andp.202070020

[25] J. Huang, H. Huo, M. Zhuang, and C. Lee, Efficient generation of spin cat states with twist-and-turn dynamics via machine optimization, Phys. Rev. A 105, 062456 (2022).
https://doi.org/10.1103/PhysRevA.105.062456

[26] J. Cai, F. Jelezko, and M. B. Plenio, Hybrid sensors based on colour centres in diamond and piezoactive layers, Nature Comm 5, 4065 (2014).
https://doi.org/10.1038/ncomms5065

[27] T. Zhang, G.-Q. Liu, W.-H. Leong, C.-F. Liu, M.-H. Kwok, T. Ngai, R.-B. Liu, and Q. Li, Hybrid nanodiamond quantum sensors enabled by volume phase transitions of hydrogels, Nat Commun 9, 3188 (2018).
https://doi.org/10.1038/s41467-018-05673-9

[28] R. Kitagawa, S. Nagata, K. Arai, K. Mizuno, T. Tsuji, I. Fujisaki, S. Urashita, T. Kohashi, Y. Takamura, T. Iwasaki, S. Nakagawa, and M. Hatano, Pressure sensor using a hybrid structure of a magnetostrictive layer and nitrogen-vacancy centers in diamond, Phys. Rev. Appl. 19, 044089 (2023).
https://doi.org/10.1103/PhysRevApplied.19.044089

[29] B. Tratzmiller, Q. Chen, I. Schwartz, S. F. Huelga, and M. B. Plenio, Limited-control metrology approaching the heisenberg limit without entanglement preparation, Phys. Rev. A 101, 032347 (2020).
https://doi.org/10.1103/PhysRevA.101.032347

[30] A. Gruber, A. Dräbenstedt, C. Tietz, L. Fleury, J. Wrachtrup, and C. von Borczyskowski, Scanning confocal optical microscopy and magnetic resonance on single defect centers, Science 276, 2012 (1997).
https://doi.org/10.1126/science.276.5321.2012

[31] Y.Wu, F. Jelezko, M. B. Plenio, and T. Weil, Diamond quantum devices in biology, Angew. Chem. Int. 55, 6586 (2016).
https://doi.org/10.1002/anie.201506556

[32] L. Zhang, A. Datta, and I. A. Walmsley, Precision metrology using weak measurements, Phys. Rev. Lett. 114, 210801 (2015).
https://doi.org/10.1103/PhysRevLett.114.210801

[33] P. Wang, C. Chen, X. Peng, J. Wrachtrup, and R.-B. Liu, Characterization of arbitrary-order correlations in quantum baths by weak measurement, Phys. Rev. Lett. 123, 050603 (2019).
https://doi.org/10.1103/PhysRevLett.123.050603

[34] M. Pfender, P. Wang, H. Sumiya, S. Onoda, W. Yang, D. B. R. Dasari, P. Neumann, X.-Y. Pan, J. Isoya, R.-B. Liu, and J. Wrachtrup, High-resolution spectroscopy of single nuclear spins via sequential weak measurements, Nature Comm 10, 594 (2019).
https://doi.org/10.1038/s41467-019-08544-z

[35] K. S. Cujia, J. M. Boss, K. Herb, and J. Z. . C. L. Degen, Tracking the precession of single nuclear spins by weak measurements, Nature 571, 230–233 (2019).
https://doi.org/10.1038/s41586-019-1334-9

[36] J. Meinel, V. Vorobyov, P. Wang, M. P. Boris Yavkin, S. O. Hitoshi Sumiya, R.-B. L. Junichi Isoya, and J. Wrachtrup, Quantum nonlinear spectroscopy of single nuclear spins, Nature Comm 13, 5318 (2022).
https://doi.org/10.1038/s41467-022-32610-8

[37] I. Schwartz, J. Rosskopf, B. T. Simon Schmitt, Q. Chen, L. P. McGuinness, F. Jelezko, and M. B. Plenio, Blueprint for nanoscale nmr, Sci Rep 9, 6938 (2019).
https://doi.org/10.1038/s41598-019-43404-2

[38] S. Schmitt, T. Gefen, F. M. Stürner, T. Unden, G. Wolff, C. Müller, J. Scheuer, B. Naydenov, M. Markham, S. Pezzagna, J. Meijer, I. Schwarz, M. B. Plenio, A. Retzker, L. P. McGuinness, and F. Jelezko, Submillihertz magnetic spectroscopy performed with a nanoscale quantum sensor, Science 356, 832 (2017).
https://doi.org/10.1126/science.aam5532

[39] J. Boss, K. Cujia, J. Zopes, and C. Degen, Quantum sensing with arbitrary frequency resolution, Science 356, 837 (2017).
https://doi.org/10.1126/science.aam7009

[40] J. Choi, A. L. Shaw, I. S. Madjarov, X. Xie, R. Finkelstein, J. P. Covey, J. S. Cotler, D. K. Mark, H.-Y. Huang, A. Kale, H. Pichler, F. G. S. L. Brandão, S. Choi, and M. Endres, Preparing random states and benchmarking with many-body quantum chaos, Nature 613, 468 (2023).
https://doi.org/10.1038/s41586-022-05442-1

[41] W. W. Ho and S. Choi, Exact emergent quantum state designs from quantum chaotic dynamics, Phys. Rev. Lett. 128, 060601 (2022).
https://doi.org/10.1103/PhysRevLett.128.060601

[42] J. S. Cotler, D. K. Mark, H.-Y. Huang, F. Hernández, J. Choi, A. L. Shaw, M. Endres, and S. Choi, Emergent quantum state designs from individual many-body wave functions, PRX Quantum 4, 010311 (2023).
https://doi.org/10.1103/PRXQuantum.4.010311

[43] M. Oszmaniec, R. Augusiak, C. Gogolin, J. Kołodyński, A. Acín, and M. Lewenstein, Random bosonic states for robust quantum metrology, Phys. Rev. X 6, 041044 (2016).
https://doi.org/10.1103/PhysRevX.6.041044

[44] P. Hyllus, O. Gühne, and A. Smerzi, Not all pure entangled states are useful for sub-shot-noise interferometry, Phys. Rev. A 82, 012337 (2010).
https://doi.org/10.1103/PhysRevA.82.012337

[45] G. Tóth and I. 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

[46] H. Cramér, Mathematical Methods of Statistics (PMS-9), Volume 9 (Princeton University Press, 1946).

[47] P. Hyllus, W. Laskowski, R. Krischek, C. Schwemmer, W. Wieczorek, H. Weinfurter, L. Pezzé, and A. Smerzi, Fisher information and multiparticle entanglement, Phys. Rev. A 85, 022321 (2012).
https://doi.org/10.1103/PhysRevA.85.022321

[48] L. Pezzé and A. Smerzi, Entanglement, nonlinear dynamics, and the heisenberg limit, Phys. Rev. Lett. 102, 100401 (2009).
https://doi.org/10.1103/PhysRevLett.102.100401

[49] D. M. Greenberger, M. A. Horne, and A. Zeilinger, Bell's Theorem, Quantum Theory and Conceptions of the Universe (Springer Netherlands, 1989).
https://doi.org/10.1007/978-94-017-0849-4_10

[50] 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

[51] J. Casanova, Z.-Y. Wang, J. F. Haase, and M. B. Plenio, Robust dynamical decoupling sequences for individual-nuclear-spin addressing, Phys. Rev. A 92, 042304 (2015).
https://doi.org/10.1103/PhysRevA.92.042304

Cited by

[1] Hai-Long Shi, Augusto Smerzi, and Luca Pezze, "Quantum chaos, randomness and universal scaling of entanglement in various Krylov spaces", SciPost Physics 19 4, 102 (2025).

The above citations are from Crossref's cited-by service (last updated successfully 2026-08-09 13:56:31) and SAO/NASA ADS (last updated successfully 2026-08-09 13:56:32). The list may be incomplete as not all publishers provide suitable and complete citation data.

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.