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  • Open Access

Efficient characterization of N-beam Gaussian fields through photon-number measurements: Quantum universal invariants

Nazarii Sudak1, Artur Barasiński1,*, Jan Peřina, Jr.2,†, and Antonín Černoch3

  • 1Institute of Theoretical Physics, University of Wroclaw, Plac Maxa Borna 9, 50-204 Wrocław, Poland
  • 2Joint Laboratory of Optics of Palacký University and Institute of Physics of CAS, Faculty of Science, Palacký University, 17. listopadu 12, 779 00 Olomouc, Czech Republic
  • 3Joint Laboratory of Optics, Institute of Physics of CAS, 17. listopadu 50a, 779 00 Olomouc, Czech Republic

  • *Contact author: artur.barasinski@uwr.edu.pl
  • †Contact author: jan.perina.jr@upol.cz
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Phys. Rev. Research 7, 023278 – Published 18 June, 2025

DOI: https://doi.org/10.1103/5hnr-6jbf

Abstract

Quantum universal invariants of general N-beam Gaussian fields are investigated from the point of view of fields' intensity moments. A method that uniquely links these invariants, including the global and marginal fields' purities, to intensity moments is suggested. Determination of these invariants identifies the Gaussian states including their quantum correlations. In particular, the Peres-Horodecki separability criterion is reformulated in terms of quantum universal invariants, and consequently in terms of experimental intensity moments, offering a practical tool for determining the entanglement or separability of these states. The approach is experimentally demonstrated by determining the invariants of noisy symmetric three-beam Gaussian states using photon-number resolved measurements. Furthermore, their entanglement properties are analyzed and characterized.

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References (53)

  1. G. Weihs, T. Jennewein, C. Simon, H. Weinfurter, and A. Zeilinger, Violation of Bell's inequality under strict Einstein locality conditions, Phys. Rev. Lett. 81, 5039 (1998).
  2. A. Zavatta, V. Parigi, M. S. Kim, H. Jeong, and M. Bellini, Experimental demonstration of the bosonic commutation relation via superpositions of quantum operations on thermal light fields, Phys. Rev. Lett. 103, 140406 (2009).
  3. J. Aasi et al., Enhanced sensitivity of the LIGO gravitational wave detector by using squeezed states of light, Nat. Photon. 7, 613 (2013).
  4. C. H. Bennet and G. Brassard, Quantum cryptography: Public key distribution and coin tossing, in Proceedings of IEEE International Conference on Computers, Systems and Signal Processing (IEEE, New York, 1984), p. 175.
  5. J. L. O'Brien, A. Furusawa, and J. Vučković, Photonic quantum technologies, Nat. Photon. 3, 687 (2009).
  6. M. A. Taylor, J. Janousek, V. Daria, J. Knittel, B. Hage, H.-A. Bachor, and W. P. Bowen, Biological measurement beyond the quantum limit, Nat. Photon. 7, 229 (2013).
  7. S.-K. Liao et al., Satellite-relayed intercontinental quantum network, Phys. Rev. Lett. 120, 030501 (2018).
  8. M. Cooper, M. Karpiński, and B. J. Smith, Local mapping of detector response for reliable quantum state estimation, Nat. Commun. 5, 4332 (2014).
  9. U. Leonhardt, Measuring the Quantum State of Light (Cambridge University Press, Cambridge, United Kingdom, 1997).
  10. A. I. Lvovsky and M. G. Raymer, Continuous-variable optical quantum-state tomography, Rev. Mod. Phys. 81, 299 (2009).
  11. O. Haderka, V. Michálek, V. Urbášek, and M. Ježek, Fast time-domain balanced homodyne detection of light, Appl. Opt. 48, 2884 (2009).
  12. J. DiGuglielmo, A. Samblowski, B. Hage, C. Pineda, J. Eisert, and R. Schnabel, Experimental unconditional preparation and detection of a continuous bound entangled state of light, Phys. Rev. Lett. 107, 240503 (2011).
  13. O. Haderka, M. Hamar, and J. Peřina, Jr., Experimental multi-photon-resolving detector using a single avalanche photodiode, Eur. Phys. J. D 28, 149 (2004).
  14. M. Avenhaus, K. Laiho, M. V. Chekhova, and C. Silberhorn, Accessing higher order correlations in quantum optical states by time multiplexing, Phys. Rev. Lett. 104, 063602 (2010).
  15. J. Sperling, M. Bohmann, W. Vogel, G. Harder, B. Brecht, V. Ansari, and C. Silberhorn, Uncovering quantum correlations with time-multiplexed click detection, Phys. Rev. Lett. 115, 023601 (2015).
  16. O. Haderka, J. Peřina, Jr., M. Hamar, and J. Peřina, Direct measurement and reconstruction of nonclassical features of twin beams generated in spontaneous parametric down-conversion, Phys. Rev. A 71, 033815 (2005).
  17. J. Peřina, Jr., M. Hamar, V. Michálek, and O. Haderka, Photon-number distributions of twin beams generated in spontaneous parametric down-conversion and measured by an intensified CCD camera, Phys. Rev. A 85, 023816 (2012).
  18. J. Peřina, Jr., V. Michálek, and O. Haderka, Noise reduction in photon counting by exploiting spatial correlations, Phys. Rev. Appl. 8, 044018 (2017).
  19. J. Peřina, Jr., O. Haderka, A. Allevi, and M. Bondani, Absolute calibration of photon-number-resolving detectors with an analog output using twin beams, Appl. Phys. Lett. 104, 041113 (2014).
  20. M. Ramilli, A. Allevi, V. Chmill, M. Bondani, M. Caccia, and A. Andreoni, Photon-number statistics with silicon photomultipliers, J. Opt. Soc. Am. B 27, 852 (2010).
  21. G. Chesi, L. Malinverno, A. Allevi, R. Santoro, M. Caccia, and M. Bondani, Measuring nonclassicality with silicon photomultipliers, Opt. Lett. 44, 1371 (2019).
  22. G. Harder, T. J. Bartley, A. E. Lita, S. W. Nam, T. Gerrits, and C. Silberhorn, Single-mode parametric-down-conversion states with 50 photons as a source for mesoscopic quantum optics, Phys. Rev. Lett. 116, 143601 (2016).
  23. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, 1995).
  24. J. Peřina, Jr., I. I. Arkhipov, V. Michálek, and O. Haderka, Non-classicality and entanglement criteria for bipartite optical fields characterized by quadratic detectors, Phys. Rev. A 96, 043845 (2017).
  25. J. Peřina, Jr., O. Haderka, and V. Michálek, Non-classicality and entanglement criteria for bipartite optical fields characterized by quadratic detectors. II. Criteria based on probabilities, Phys. Rev. A 102, 043713 (2020).
  26. J. Peřina, Jr., P. Pavlíček, V. Michálek, R. Machulka, and O. Haderka, Nonclassicality criteria for N-dimensional optical fields detected by quadratic detectors, Phys. Rev. A 105, 013706 (2022).
  27. A. Barasiński, J. Peřina, and A. Černoch, Quantification of quantum correlations in two-beam Gaussian states using photon-number measurements, Phys. Rev. Lett. 130, 043603 (2023).
  28. A. V. Dodonov, V. V. Dodonov, and S. S. Mizrahi, Separability dynamics of two-mode Gaussian states in parametric conversion and amplification, J. Phys. A: Math. Gen. 38, 683 (2005).
  29. A. Serafini, Multimode uncertainty relations and separability of continuous variable states, Phys. Rev. Lett. 96, 110402 (2006).
  30. J. Williamson, On the algebraic problem concerning the normal forms of linear dynamical systems, Am. J. Math. 58, 141 (1936).
  31. R. Simon, S. Chaturvedi, and V. Srinivasan, Congruences and canonical forms for a positive matrix: Application to the Schweinler–Wigner extremum principle, J. Math. Phys. 40, 3632 (1999).
  32. I. Kogias, A. R. Lee, S. Ragy, and G. Adesso, Quantification of Gaussian quantum steering, Phys. Rev. Lett. 114, 060403 (2015).
  33. J. Peřina, Quantum Statistics of Linear and Nonlinear Optical Phenomena (Kluwer, Dordrecht, 1991).
  34. G. Adesso and F. Illuminati, Entanglement in continuous-variable systems: Recent advances and current perspectives, J. Phys. A 40, 7821 (2007).
  35. C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, Gaussian quantum information, Rev. Mod. Phys. 84, 621 (2012).
  36. H. P. Robertson, The uncertainty principle, Phys. Rev. 34, 163 (1929).
  37. E. Schrödinger, Zum Heisenbergschen unschärfeprinzip, Ber. Kgl. Akad. Wiss. Berlin 24, 296 (1930) [Bulg. J. Phys. 26, 193 (1999)].
  38. A. Serafini, Detecting entanglement by symplectic uncertainty relations, J. Opt. Soc. Am. B 24, 347 (2007).
  39. I. S. Gradshtein and I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed. (Academic, San Diego, 2000).
  40. J. Peřina, Jr., Waves in spatio-spectral and -temporal coherence of evolving ultra-intense twin beams, Sci. Rep. 9, 4256 (2019).
  41. J. Peřina, Jr., A. Černoch, and J. Soubusta, Compound twin beams without the need of genuine photon-number-resolving detection, Phys. Rev. Appl. 16, 024061 (2021).
  42. A. P. Dempster, N. M. Laird, and D. B. Rubin, Maximum likelihood from incomplete data via the EM algorithm, J. R. Statist. Soc. B 39, 1 (1977).
  43. Y. Vardi and D. Lee, From image deblurring to optimal investments: Maximum likelihood solutions for positive linear inverse problems, J. R. Statist. Soc. B 55, 569 (1993).
  44. C. T. Lee, Measure of the nonclassicality of nonclassical states, Phys. Rev. A 44, R2775 (1991).
  45. J. Peřina, Jr., N. Sudak, and A. Barasiński, and A. Černoch, Quantum correlations in three-beam symmetric Gaussian states accessed via photon-number-resolving detection and quantum universal invariants, arXiv:2505.11303.
  46. G. Adesso, A. Serafini, and F. Illuminati, Extremal entanglement and mixedness in continuous variable systems, Phys. Rev. A 70, 022318 (2004).
  47. V. Michálek, J. Peřina, Jr., and O. Haderka, Experimental quantification of the entanglement of noisy twin beams, Phys. Rev. Appl. 14, 024003 (2020).
  48. K. Thapliyal, J. Peřina, Jr., V. Michálek, O. Haderka, and R. Machulka, Experimental photon addition and subtraction in multi-mode and entangled optical fields, Opt. Lett. 49, 4521 (2024).
  49. A. Peres, Separability criterion for density matrices, Phys. Rev. Lett. 77, 1413 (1996).
  50. P. Horodecki, Separability criterion and inseparable mixed states with positive partial transposition, Phys. Lett. A 232, 333 (1997).
  51. N. Sudak, A. Barasiński, J. Peřina, Jr., and A. Černoch, Efficient characterization of N-beam Gaussian fields through photon-number measurements: Quantum universal invariants, Zenodo (2025), https://doi.org/10.5281/zenodo.15039026.
  52. J. Peřina, Jr., O. Haderka, M. Hamar, and V. Michálek, Absolute detector calibration using twin beams, Opt. Lett. 37, 2475 (2012).
  53. J. Peřina, Jr., O. Haderka, V. Michálek, and M. Hamar, State reconstruction of a multimode twin beam using photodetection, Phys. Rev. A 87, 022108 (2013).

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Phys. Rev. Research 7, 023278– Published 18 June, 2025
  • Received 6 January 2025
  • Accepted 27 May 2025
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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.

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