- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Research

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

Generalized squeezing as a witness of various quantum properties

Šimon Bräuer*, Tomáš Opatrný, and Petr Marek

  • Department of Optics, Faculty of Science, Palacký University, 17, Listopadu 12, 77146 Olomouc, Czech Republic

  • *Contact author: brauer@optics.upol.cz
PDF

Phys. Rev. Research 7, 033176 – Published 21 August, 2025

DOI: https://doi.org/10.1103/9cxn-t1vf

Abstract

Quantum systems can be prepared in an infinite continuum of states, but only some of them can be used as resources for quantum technologies. Discerning whether a specific quantum state falls into this class is often a challenging task. We show that it can be performed by looking at the squeezing of the quantum states—a scenario in which the variance of some observable operator is suppressed below the threshold given by the nonuseful states. We discuss the general concept first; then, we illustrate the approach by evaluating cubic nonlinear squeezing in the system of collective atomic spins.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (76)

  1. S. Konno, W. Asavanant, F. Hanamura, H. Nagayoshi, K. Fukui, A. Sakaguchi, R. Ide, F. China, M. Yabuno, S. Miki, H. Terai, K. Takase, M. Endo, P. Marek, R. Filip, P. van Loock, and A. Furusawa, Logical states for fault-tolerant quantum computation with propagating light, Science 383, 289 (2024).
  2. P. Marek, Ground state nature and nonlinear squeezing of Gottesman-Kitaev-Preskill states, Phys. Rev. Lett. 132, 210601 (2024).
  3. D. Gottesman, A. Kitaev, and J. Preskill, Encoding a qubit in an oscillator, Phys. Rev. A 64, 012310 (2001).
  4. J. E. Bourassa, R. N. Alexander, M. Vasmer, A. Patil, I. Tzitrin, T. Matsuura, D. Su, B. Q. Baragiola, S. Guha, G. Dauphinais, K. K. Sabapathy, N. C. Menicucci, and I. Dhand, Blueprint for a scalable photonic fault-tolerant quantum computer, Quantum 5, 392 (2021).
  5. B. Q. Baragiola, G. Pantaleoni, R. N. Alexander, A. Karanjai, and N. C. Menicucci, All-Gaussian universality and fault tolerance with the Gottesman-Kitaev-Preskill code, Phys. Rev. Lett. 123, 200502 (2019).
  6. A. L. Grimsmo, J. Combes, and B. Q. Baragiola, Quantum computing with rotation-symmetric bosonic codes, Phys. Rev. X 10, 011058 (2020).
  7. A. L. Grimsmo and S. Puri, Quantum error correction with the Gottesman-Kitaev-Preskill code, PRX Quantum 2, 020101 (2021).
  8. B. M. Terhal, J. Conrad, and C. Vuillot, Towards scalable bosonic quantum error correction, Quantum Sci. Technol. 5, 043001 (2020).
  9. V. Giovannetti, S. Lloyd, and L. Maccone, Quantum-enhanced measurements: Beating the standard quantum limit, Science 306, 1330 (2004).
  10. V. Giovannetti, S. Lloyd, and L. Maccone, Quantum metrology, Phys. Rev. Lett. 96, 010401 (2006).
  11. R. Schnabel, N. Mavalvala, D. E. McClelland, and P. K. Lam, Quantum metrology for gravitational wave astronomy, Nat. Commun. 1, 121 (2010).
  12. A. Cao, W. J. Eckner, T. Lukin Yelin, A. W. Young, S. Jandura, L. Yan, K. Kim, G. Pupillo, J. Ye, N. Darkwah Oppong, and A. M. Kaufman, Multi-qubit gates and Schrödinger cat states in an optical clock, Nature (London) 634, 315 (2024).
  13. D. DeMille, N. R. Hutzler, A. M. Rey, and T. Zelevinsky, Quantum sensing and metrology for fundamental physics with molecules, Nat. Phys. 20, 741 (2024).
  14. Y. L. Len, T. Gefen, A. Retzker, and J. Kołodyński, Quantum metrology with imperfect measurements, Nat. Commun. 13, 6971 (2022).
  15. D. Lee, W. Shin, S. Park, J. Kim, and H. Shin, NOON-state interference in the frequency domain, Light Sci. Appl. 13, 90 (2024).
  16. G. A. Brawley, M. R. Vanner, P. E. Larsen, S. Schmid, A. Boisen, and W. P. Bowen, Nonlinear optomechanical measurement of mechanical motion, Nat. Commun. 7, 10988 (2016).
  17. M. Gessner, A. Smerzi, and L. Pezzè, Metrological nonlinear squeezing parameter, Phys. Rev. Lett. 122, 090503 (2019).
  18. C. Oh, K. Park, R. Filip, H. Jeong, and P. Marek, Optical estimation of unitary Gaussian processes without phase reference using Fock states, New J. Phys. 22, 123039 (2020).
  19. W. Jia, V. Xu, K. Kuns, M. Nakano, L. Barsotti, M. Evans, N. Mavalvala, M. Of the Ligo Scientific Collaboration., R. Abbott, I. Abouelfettouh, R. X. Adhikari, A. Ananyeva, S. Appert, K. Arai, N. Aritomi, S. M. Aston, M. Ball, S. W. Ballmer, D. Barker, B. K. Berger et al., Squeezing the quantum noise of a gravitational-wave detector below the standard quantum limit, Science 385, 1318 (2024).
  20. D. F. Walls, Squeezed states of light, Nature (London) 306, 141 (1983).
  21. R. Filip, P. Marek, and U. L. Andersen, Measurement-induced continuous-variable quantum interactions, Phys. Rev. A 71, 042308 (2005).
  22. Y. Miwa, J.-i. Yoshikawa, N. Iwata, M. Endo, P. Marek, R. Filip, P. van Loock, and A. Furusawa, Exploring a new regime for processing optical qubits: Squeezing and unsqueezing single photons, Phys. Rev. Lett. 113, 013601 (2014).
  23. T. Serikawa, J.-I. Yoshikawa, S. Takeda, H. Yonezawa, T. C. Ralph, E. H. Huntington, and A. Furusawa, Generation of a cat state in an optical sideband, Phys. Rev. Lett. 121, 143602 (2018).
  24. K. Huang, H. Le Jeannic, J. Ruaudel, V. B. Verma, M. D. Shaw, F. Marsili, S. W. Nam, E. Wu, H. Zeng, Y.-C. Jeong, R. Filip, O. Morin, and J. Laurat, Optical synthesis of large-amplitude squeezed coherent-state superpositions with minimal resources, Phys. Rev. Lett. 115, 023602 (2015).
  25. M. H. Michael, M. Silveri, R. T. Brierle, V. V. Albert, J. Salmilehto, L. Jiang, and S. M. Girvin, New class of quantum error-correcting codes for a bosonic mode, Phys. Rev. X 6, 031006 (2016).
  26. K. Miyata, H. Ogawa, P. Marek, R. Filip, H. Yonezawa, J.-I. Yoshikawa, and A. Furusawa, Implementation of a quantum cubic gate by an adaptive non-Gaussian measurement, Phys. Rev. A 93, 022301 (2016).
  27. P. Marek, R. Filip, H. Ogawa, A. Sakaguchi, S. Takeda, J.-I. Yoshikawa, and A. Furusawa., General implementation of arbitrary nonlinear quadrature phase gates, Phys. Rev. A 97, 022329 (2018).
  28. P. Marek, R. Filip, and A. Furusawa, Deterministic implementation of weak quantum cubic nonlinearity, Phys. Rev. A 84, 053802 (2011).
  29. A. Sakaguchi, S. Konno, F. Hanamura, W. Asavanant, K. Takase, H. Ogawa, P. Marek, R. Filip, J.-I. Yoshikawa, E. Huntington, H. Yonezawa, and A. Furusawa, Nonlinear feedforward enabling quantum computation, Nat. Commun. 14, 3817 (2023).
  30. J. Aasi, J. Abadie, B. P. Abbott, R. Abbott, T. D. Abbott, M. R. Abernathy, C. Adams, T. Adams, P. Addesso, R. X. Adhikari, C. Affeldt, O. D. Aguiar, P. Ajith, B. Allen, E. Amador Ceron, D. Amariutei, S. B. Anderson, W. G. Anderson, K. Arai, M. C. Araya et al., Enhanced sensitivity of the LIGO gravitational wave detector by using squeezed states of light, Nat. Photon. 7, 613 (2013).
  31. F. Acernese, M. Agathos, L. Aiello, A. Allocca, A. Amato, S. Ansoldi, S. Antier, M. Arène, N. Arnaud, S. Ascenzi, P. Astone, F. Aubin, S. Babak, P. Bacon, F. Badaracco, M. K. M. Bader, J. Baird, F. Baldaccini, G. Ballardin et al. (Virgo Collaboration), Increasing the astrophysical reach of the advanced virgo detector via the application of squeezed vacuum states of light, Phys. Rev. Lett. 123, 231108 (2019).
  32. 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).
  33. X. Deng, S. Li, Z.-J. Chen, Z. Ni, Y. Cai, J. Mai, L. Zhang, P. Zheng, H. Yu, C.-L. Zou, S. Liu, F. Yan, Y. Xu, and D. Yu, Quantum-enhanced metrology with large Fock states, Nat. Phys. 20, 1874 (2024).
  34. J.P. Dowling, Quantum optical metrology—The lowdown on high-N00N states, Contemp. Phys. 49, 125 (2008).
  35. K. C. McCormick, J. Keller, S. C. Burd, D. J. Wineland, A. C. Wilson, and D. Leibfried, Quantum-enhanced sensing of a single-ion mechanical oscillator, Nature (London) 572, 86 (2019).
  36. Š. Bräuer and P. Marek, Generation of quantum states with nonlinear squeezing by Kerr nonlinearity, Opt. Express 29, 22648 (2021).
  37. V. Kala, R. Filip, and P. Marek, Cubic nonlinear squeezing and its decoherence, Opt. Express 30, 31456 (2022).
  38. V. Kala, M. Walschaers, R. Filip, and P. Marek, Non-Gaussian state teleportation with a nonlinear feedforward, arXiv:2404.06438.
  39. F. Albarelli, M. G. Genoni, M. G. A. Paris, and A. Ferraro, Resource theory of quantum non-Gaussianity and Wigner negativity, Phys. Rev. A 98, 052350 (2018).
  40. E. Chitambar and G. Gour, Quantum resource theories, Rev. Mod. Phys. 91, 025001 (2019).
  41. A. S. Sørensen and K. Mølmer, Entanglement and extreme spin squeezing, Phys. Rev. Lett. 86, 4431 (2001).
  42. L. Lachman and R. Filip, Quantum non-Gaussianity of light and atoms, Prog. Quantum Electron. 83, 100395 (2022).
  43. M. Walschaers, Non-Gaussian quantum states and where to find them, PRX Quantum 2, 030204 (2021).
  44. S. Lloyd and S. L. Braunstein, Quantum computation over continuous variables, Phys. Rev. Lett. 82, 1784 (1999).
  45. H. Strobel, W. Muessel, D. Linnemann, T. Zibold, D. B. Hume, L. Pezzè, A. Smerzi, and M. K. Oberthaler, Fisher information and entanglement of non-Gaussian spin states, Science 345, 424 (2014).
  46. 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).
  47. A. Evrard, V. Makhalov, T. Chalopin, L. A. Sidorenkov, J. Dalibard, R. Lopes, and S. Nascimbene, Enhanced magnetic sensitivity with non-Gaussian quantum fluctuations, Phys. Rev. Lett. 122, 173601 (2019).
  48. T. Opatrný, H. Saberi, E. Brion, and K. Mølmer, Counterdiabatic driving in spin squeezing and Dicke-state preparation, Phys. Rev. A 93, 023815 (2016).
  49. K. Park, P. Marek, and R. Filip, Deterministic nonlinear phase gates induced by a single qubit, New J. Phys. 20, 053022 (2018).
  50. S. Ghose and B. C. Sanders, Non-Gaussian ancilla states for continuous variable quantum computation via Gaussian maps, J. Mod. Opt. 54, 855 (2007).
  51. S. Konno, A. Sakaguchi, W. Asavanant, H. Ogawa, M. Kobayashi, P. Marek, R. Filip, J.-I. Yoshikawa, and A. Furusawa, Nonlinear squeezing for measurement-based non-Gaussian operations in time domain, Phys. Rev. Appl. 15, 024024 (2021).
  52. R. Yanagimoto, T. Onodera, E. Ng, L. G. Wright, P. L. McMahon, and H. Mabuchi, Engineering a Kerr-based deterministic cubic phase gate via Gaussian operations, Phys. Rev. Lett. 124, 240503 (2020).
  53. M. Yukawa, K. Miyata, H. Yonezawa, P. Marek, R. Filip, and A. Furusawa, Emulating quantum cubic nonlinearity, Phys. Rev. A 88, 053816 (2013).
  54. U. Chabaud, D. Markham, and F. Grosshans, Stellar representation of non-Gaussian quantum states, Phys. Rev. Lett. 124, 063605 (2020).
  55. J. Fiurášek, Efficient construction of witnesses of the stellar rank of nonclassical states of light, Opt. Express 30, 30630 (2022).
  56. U. Chabaud, G. Roeland, M. Walschaers, F. Grosshans, V. Parigi, D. Markham, and N. Treps, Certification of non-Gaussian states with operational measurements, PRX Quantum 2, 020333 (2021).
  57. S. L. Braunstein and C. M. Caves, Statistical distance and the geometry of quantum states, Phys. Rev. Lett. 72, 3439 (1994).
  58. R. Jozsa, Fidelity for mixed quantum states, J. Mod. Opt. 41, 2315 (1994).
  59. L.-M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, Inseparability criterion for continuous variable systems, Phys. Rev. Lett. 84, 2722 (2000).
  60. J. Ma, X. Wang, C. P. Sun, and F. Nori, Quantum spin squeezing, Phys. Rep. 509, 89 (2011).
  61. 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).
  62. M. Kitagawa and M. Ueda, Squeezed spin states, Phys. Rev. A 47, 5138 (1993).
  63. J. A. Hines, S. V. Rajagopal, G. L. Moreau, M. D. Wahrman, N. A. Lewis, O. Marković, and M. Schleier-Smith, Spin squeezing by Rydberg dressing in an array of atomic ensembles, Phys. Rev. Lett. 131, 063401 (2023).
  64. 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).
  65. H. Zhou, J. Choi, S. Choi, R. Landig, A. M. Douglas, J. Isoya, F. Jelezko, S. Onoda, H. Sumiya, P. Cappellaro, H. S. Knowles, H. Park, and M. D. Lukin, Quantum metrology with strongly interacting spin systems, Phys. Rev. X 10, 031003 (2020).
  66. V. Montenegro, Heisenberg-limited spin-mechanical gravimetry, Phys. Rev. Res. 7, 013016 (2025).
  67. 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 (London) 603, 604 (2022).
  68. S. L. Braunstein, Squeezing as an irreducible resource, Phys. Rev. A 71, 055801 (2005).
  69. B. Yadin, F. C. Binder, J. Thompson, V. Narasimhachar, M. Gu, and M. S. Kim, Operational resource theory of continuous-variable nonclassicality, Phys. Rev. X 8, 041038 (2018).
  70. G. Carrara, M. G. Genoni, S. Cialdi, M. G. A. Paris, and S. Olivares, Squeezing as a resource to counteract phase diffusion in optical phase estimation, Phys. Rev. A 102, 062610 (2020).
  71. S. Colombo, E. Pedrozo-Peñafiel, A. F. Adiyatullin, Z. Li, E. Mendez, C. Shu, and V. Vuletic, Time-reversal-based quantum metrology with many-body entangled states, Nat. Phys. 18, 925 (2022).
  72. W. Muessel, H. Strobel, D. Linnemann, T. Zibold, B. Juliá-Díaz, and M. K. Oberthaler, Twist-and-turn spin squeezing in Bose-Einstein condensates, Phys. Rev. A 92, 023603 (2015).
  73. T. Holstein and H. Primakoff, Field dependence of the intrinsic domain magnetization of a ferromagnet, Phys. Rev. 58, 1098 (1940).
  74. D. W. Moore, A. A. Rakhubovsky, and R. Filip, Estimation of squeezing in a nonlinear quadrature of a mechanical oscillator, New J. Phys. 21, 113050 (2019).
  75. V. Kala, D. Kopylov, P. Marek, and P. Sharapova, Nonlinear squeezing generation via multimode PDC and single photon measurement, Opt. Express 33, 14000 (2025).
  76. Š. Bräuer, T. Opatrný, and P. Marek, Generalized squeezing as a witness of various quantum properties, Zenodo (2025), https://doi.org/10.5281/zenodo.16034698.

Outline

Information

Phys. Rev. Research 7, 033176– Published 21 August, 2025
  • Received 19 February 2025
  • Accepted 14 July 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