- Physical Review Journals
All JournalsPhysics Magazine

Physical Review A

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

Experimental evidence-based sub-Rayleigh source discrimination

Saurabh U. Shringarpure1,*, Yong Siah Teo1,2,†, Hyunseok Jeong1,‡, Michael Evans3,§, Luis L. Sánchez-Soto4,5,6, Antonin Grateau7, Alexander Boeschoten7, and Nicolas Treps7

  • *Contact author: saurabh.s@snu.ac.kr
  • †Contact author: ys_teo@sejong.ac.kr
  • ‡Contact author: h.jeong37@gmail.com
  • §Contact author: mevansthree.evans@utoronto.ca
PDF

Phys. Rev. A 114, 043713 – Published 13 October, 2026

DOI: https://doi.org/10.1103/9ffd-pb96

Abstract

We propose a Bayesian evidence-based inference framework built on relative belief ratios and apply it to discriminating between one and two incoherent optical point sources using spatial-mode demultiplexing (SPADE). Unlike the Helstrom measurement, SPADE requires no collective detection and is optimal for asymptotically large samples. Our method avoids ad hoc statistical constructs and draws inference directly from the information contained in the data, with all assumptions entering only through the likelihood model and prior beliefs. Using experimental evidence, we demonstrate the superior resolving performance of SPADE over direct imaging from a perspective that generalizes to multiple sources and offers a practical, robust approach to analyzing quantum-enhanced superresolution.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (66)

  1. C. W. Helstrom, Quantum Detection and Estimation Theory (Academic Press, New York, 1976).
  2. M. Hayashi, Quantum Information Theory: Mathematical Foundation (Springer, Berlin, 2017).
  3. M. M. Wilde, Quantum Information Theory (Cambridge University Press, Cambridge, 2017).
  4. C. W. Helstrom, Quantum detection and estimation theory, J. Stat. Phys. 1, 231 (1969).
  5. A. S. Holevo, Statistical decision theory for quantum systems, J. Multivar. Anal. 3, 337 (1973).
  6. M. Ban, K. Kurokawa, R. Momose, and O. Hirota, Optimum measurements for discrimination among symmetric quantum states and parameter estimation, Int. J. Theor. Phys. 36, 1269 (1997).
  7. S. M. Barnett and S. Croke, Quantum state discrimination, Adv. Opt. Photonics 1, 238 (2009).
  8. S. Croke, E. Andersson, S. M. Barnett, C. R. Gilson, and J. Jeffers, Maximum confidence quantum measurements, Phys. Rev. Lett. 96, 070401 (2006).
  9. A. Chefles, Unambiguous discrimination between linearly independent quantum states, Phys. Lett. A 239, 339 (1998).
  10. T. Rudolph, R. W. Spekkens, and P. S. Turner, Unambiguous discrimination of mixed states, Phys. Rev. A 68, 010301(R) (2003).
  11. I. D. Ivanovic, How to differentiate between non-orthogonal states, Phys. Lett. A 123, 257 (1987).
  12. D. Dieks, Overlap and distinguishability of quantum states, Phys. Lett. A 126, 303 (1988).
  13. A. Peres, How to differentiate between non-orthogonal states, Phys. Lett. A 128, 19 (1988).
  14. J. A. Bergou, Discrimination of quantum states, J. Mod. Opt. 57, 160 (2010).
  15. J. A. Bergou, U. Futschik, and E. Feldman, Optimal unambiguous discrimination of pure quantum states, Phys. Rev. Lett. 108, 250502 (2012).
  16. J. L. Harris, Resolving power and decision theory, J. Opt. Soc. Am. 54, 606 (1964).
  17. C. Helstrom, Resolution of point sources of light as analyzed by quantum detection theory, IEEE Trans. Inf. Theory 19, 389 (1973).
  18. D. Cunningham, R. Laramore, and E. Barrett, Detection in image dependent noise, IEEE Trans. Inf. Theory 22, 603 (1976).
  19. C. O. Acuna and J. Horowitz, A statistical approach to the resolution of point sources, J. Appl. Stat. 24, 421 (1997).
  20. M. Shahram and P. Milanfar, Statistical and information-theoretic analysis of resolution in imaging, IEEE Trans. Inf. Theory 52, 3411 (2006).
  21. M. Tsang, Resolving starlight: A quantum perspective, Contemp. Phys. 60, 279 (2019).
  22. Z. Huang and C. Lupo, Quantum hypothesis testing for exoplanet detection, Phys. Rev. Lett. 127, 130502 (2021).
  23. S.-H. Lee, J. Y. Shin, A. Lee, and C. Bustamante, Counting single photoactivatable fluorescent molecules by photoactivated localization microscopy (PALM), Proc. Natl. Acad. Sci. USA 109, 17436 (2012).
  24. X. Nan, E. A. Collisson, S. Lewis, J. Huang, T. M. Tamgüney, J. T. Liphardt, F. McCormick, J. W. Gray, and S. Chu, Single-molecule superresolution imaging allows quantitative analysis of RAF multimer formation and signaling, Proc. Natl. Acad. Sci. USA 110, 18519 (2013).
  25. M. P. Backlund, Y. Shechtman, and R. L. Walsworth, Fundamental precision bounds for three-dimensional optical localization microscopy with Poisson statistics, Phys. Rev. Lett. 121, 023904 (2018).
  26. J. W. Goodman, Introduction to Fourier Optics (Roberts, Englewood, NJ, 2004).
  27. A. J. den Dekker and A. van den Bos, Resolution: A survey, J. Opt. Soc. Am. A 14, 547 (1997).
  28. V. Delaubert, N. Treps, M. Lassen, C. C. Harb, C. Fabre, P. K. Lam, and H. A. Bachor, TEM10 homodyne detection as an optimal small-displacement and tilt-measurement scheme, Phys. Rev. A 74, 053823 (2006).
  29. M. Tsang, R. Nair, and X.-M. Lu, Quantum theory of superresolution for two incoherent optical point sources, Phys. Rev. X 6, 031033 (2016).
  30. R. Nair and M. Tsang, Far-field superresolution of thermal electromagnetic sources at the quantum limit, Phys. Rev. Lett. 117, 190801 (2016).
  31. S. Z. Ang, R. Nair, and M. Tsang, Quantum limit for two-dimensional resolution of two incoherent optical point sources, Phys. Rev. A 95, 063847 (2017).
  32. M. Paúr, B. Stoklasa, Z. Hradil, L. L. Sánchez-Soto, and J. Rehacek, Achieving the ultimate optical resolution, Optica 3, 1144 (2016).
  33. F. Yang, A. Tashchilina, E. S. Moiseev, C. Simon, and A. I. Lvovsky, Far-field linear optical superresolution via heterodyne detection in a higher-order local oscillator mode, Optica 3, 1148 (2016).
  34. W.-K. Tham, H. Ferretti, and A. M. Steinberg, Beating Rayleigh's curse by imaging using phase information, Phys. Rev. Lett. 118, 070801 (2017).
  35. J. Rehacek, M. Paúr, B. Stoklasa, Z. Hradil, and L. L. Sánchez-Soto, Optimal measurements for resolution beyond the Rayleigh limit, Opt. Lett. 42, 231 (2017).
  36. U. Zanforlin, C. Lupo, P. W. R. Connolly, P. Kok, G. S. Buller, and Z. Huang, Optical quantum super-resolution imaging and hypothesis testing, Nat. Commun. 13, 5373 (2022).
  37. M. R. Grace and S. Guha, Identifying objects at the quantum limit for superresolution imaging, Phys. Rev. Lett. 129, 180502 (2022).
  38. K. Schlichtholz, T. Linowski, M. Walschaers, N. Treps, Ł. Rudnicki, and G. Sorelli, Practical tests for sub-Rayleigh source discriminations with imperfect demultiplexers, Opt. Quantum 2, 29 (2024).
  39. S. A. Wadood, K. R. Sethuraj, K. Liang, M. R. Grace, G. La Rue, S. Guha, and A. N. Vamivakas, Experimental demonstration of quantum-inspired optical symmetric hypothesis testing, Opt. Lett. 49, 750 (2024).
  40. G. Labroille, B. Denolle, P. Jian, P. Genevaux, N. Treps, and J.-F. Morizur, Efficient and mode selective spatial mode multiplexer based on multi-plane light conversion, Opt. Express 22, 15599 (2014).
  41. X.-M. Lu, H. Krovi, R. Nair, S. Guha, and J. H. Shapiro, Quantum-optimal detection of one-versus-two incoherent optical sources with arbitrary separation, npj Quantum Inf. 4, 64 (2018).
  42. R. Nichols, T. R. Bromley, L. A. Correa, and G. Adesso, Practical quantum metrology in noisy environments, Phys. Rev. A 94, 042101 (2016).
  43. M. R. Grace, Z. Dutton, A. Ashok, and S. Guha, Approaching quantum-limited imaging resolution without prior knowledge of the object location, J. Opt. Soc. Am. A 37, 1288 (2020).
  44. C. Lupo, Subwavelength quantum imaging with noisy detectors, Phys. Rev. A 101, 022323 (2020).
  45. Y. L. Len, C. Datta, M. Parniak, and K. Banaszek, Resolution limits of spatial mode demultiplexing with noisy detection, Int. J. Quantum Inf. 18, 1941015 (2020).
  46. C. Oh, S. Zhou, Y. Wong, and L. Jiang, Quantum limits of superresolution in a noisy environment, Phys. Rev. Lett. 126, 120502 (2021).
  47. G. Sorelli, M. Gessner, M. Walschaers, and N. Treps, Optimal observables and estimators for practical superresolution imaging, Phys. Rev. Lett. 127, 123604 (2021).
  48. J. O. de Almeida, J. Kołodyński, C. Hirche, M. Lewenstein, and M. Skotiniotis, Discrimination and estimation of incoherent sources under misalignment, Phys. Rev. A 103, 022406 (2021).
  49. M. Gessner, C. Fabre, and N. Treps, Superresolution limits from measurement crosstalk, Phys. Rev. Lett. 125, 100501 (2020).
  50. G. Sorelli, M. Gessner, M. Walschaers, and N. Treps, Moment-based superresolution: Formalism and applications, Phys. Rev. A 104, 033515 (2021).
  51. T. Linowski, K. Schlichtholz, G. Sorelli, M. Gessner, M. Walschaers, N. Treps, and Ł. Rudnicki, Application range of crosstalk-affected spatial demultiplexing for resolving separations between unbalanced sources, New J. Phys. 25, 103050 (2023).
  52. M. Evans, Measuring Statistical Evidence Using Relative Belief (Chapman and Hall, New York, 2015).
  53. M. Evans, Measuring statistical evidence using relative belief, Comput. Struct. Biotechnol. J. 14, 91 (2016).
  54. M. Evans and J. Tomal, Measuring statistical evidence and multiple testing, FACETS 3, 563 (2018).
  55. J. Shang, H. K. Ng, A. Sehrawat, X. Li, and B.-G. Englert, Optimal error regions for quantum state estimation, New J. Phys. 15, 123026 (2013).
  56. X. Li, J. Shang, H. K. Ng, and B.-G. Englert, Optimal error intervals for properties of the quantum state, Phys. Rev. A 94, 062112 (2016).
  57. Y. S. Teo, S. U. Shringarpure, H. Jeong, N. Prasannan, B. Brecht, C. Silberhorn, M. Evans, D. Mogilevtsev, and L. L. Sánchez-Soto, Evidence-based certification of quantum dimensions, Phys. Rev. Lett. 133, 050204 (2024).
  58. Y. S. Teo, S. U. Shringarpure, H. Jeong, N. Prasannan, B. Brecht, C. Silberhorn, M. Evans, D. Mogilevtsev, and L. L. Sánchez-Soto, Relative-belief inference in quantum information theory, Phys. Rev. A 110, 012231 (2024).
  59. P. Agarwal, N. Ali, C. Polvara, M.-I. Trappe, B.-G. Englert, and M. Hillery, State learning from pairs of states, Phys. Rev. A 111, 062428 (2025).
  60. B.-G. Englert, Lectures on Quantum State Estimation (World Scientific, Singapore, 2025).
  61. S. M. Kay, Fundamentals of Statistical Signal Processing (Prentice Hall, Upper Saddle River, NJ, 1993), Vol. 1.
  62. D. J. Benjamin et al., Redefine statistical significance, Nat. Hum. Behav. 2, 6 (2018).
  63. R. L. Wasserstein and N. A. Lazar, The ASA statement on p-values: Context, process, and purpose, Am. Stat. 70, 129 (2016).
  64. A. Grateau, A. Boeschoten, T. Favin-Lévêque, I. Herrera, and N. Treps, Multiparameter estimation for the superresolution of two incoherent sources, arXiv:2601.14876.
  65. S. Ram, E. S. Ward, and R. J. Ober, Beyond Rayleigh's criterion: A resolution measure with application to single-molecule microscopy, Proc. Natl. Acad. Sci. USA 103, 4457 (2006).
  66. S. Zhou and L. Jiang, Modern description of Rayleigh's criterion, Phys. Rev. A 99, 013808 (2019).

Outline

Information

Phys. Rev. A 114, 043713– Published 13 October, 2026
  • Received 5 August 2025
  • Revised 1 June 2026
  • Accepted 18 September 2026
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. Open access publication funded by Max Planck Society.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation