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  • Open Access

Diffusion minimization via optimal smearing in collapse and hybrid classical-quantum gravitational models

Nicolò Piccione*

  • *Contact author: nicolo.piccione@units.it
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Phys. Rev. A 113, 062211 – Published 9 June, 2026

DOI: https://doi.org/10.1103/h74z-zvfv

Abstract

Spontaneous diffusion (i.e., nonconservation of energy) is a prominent, testable prediction of collapse and hybrid classical-quantum gravitational models. Without smearing of the mass density operator, the associated heating (or energy increase) rate diverges, yet the smearing distribution is arbitrary and, on scales much larger than the smearing length rC, much of the phenomenology is expected to be insensitive to this choice. We propose to resolve this arbitrariness as follows: for a fixed rC, select the distribution that minimizes the heating rate. Conceptually, this should identify the minimal deviation from standard quantum mechanics and provide models that, once experimentally refuted, would strongly disfavor all variants with different distributions. We apply this approach to the most investigated collapse models: GRW (for Ghirardi-Rimini-Weber), CSL (for continuous spontaneous localization), and DP (for Diósi-Penrose). Notably, the Gaussian is optimal only for the GRW case. Finally, we apply it to the Tilloy-Diósi hybrid classical-quantum model of Newtonian gravity, leading to its minimally deviating variant. This version of the model is entirely determined by only one free parameter rC, and, if experimentally refuted, would strongly disfavor any other version of it.

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References (64)

  1. J. S. Bell, Speakable and Unspeakable in Quantum Mechanics: Collected Papers on Quantum Philosophy (Cambridge University Press, 2004).
  2. T. Norsen, Foundations of Quantum Mechanics (Springer, 2017).
  3. D. Dürr and D. Lazarovici, Understanding Quantum Mechanics (Springer, 2020).
  4. R. Tumulka, Foundations of Quantum Mechanics (Springer Nature, 2022), Vol. 1003.
  5. C. Kiefer, Quantum gravity–An unfinished revolution (2023), Invited contribution to EPS Grand Challenges: Physics for Society at the Horizon 2050, arXiv:2302.13047.
  6. A. Bassi and G. Ghirardi, Dynamical reduction models, Phys. Rep. 379, 257 (2003).
  7. A. Bassi, K. Lochan, S. Satin, T. P. Singh, and H. Ulbricht, Models of wave-function collapse, underlying theories, and experimental tests, Rev. Mod. Phys. 85, 471 (2013).
  8. A. Bassi, M. Dorato, and H. Ulbricht, Collapse models: A theoretical, experimental and philosophical review, Entropy 25, 645 (2023).
  9. D. Kafri, J. M. Taylor, and G. J. Milburn, A classical channel model for gravitational decoherence, New J. Phys. 16, 065020 (2014).
  10. D. Kafri, G. J. Milburn, and J. M. Taylor, Bounds on quantum communication via Newtonian gravity, New J. Phys. 17, 015006 (2015).
  11. A. Tilloy and L. Diósi, Sourcing semiclassical gravity from spontaneously localized quantum matter, Phys. Rev. D 93, 024026 (2016).
  12. A. Tilloy and L. Diósi, Principle of least decoherence for Newtonian semiclassical gravity, Phys. Rev. D 96, 104045 (2017).
  13. J. L. Gaona-Reyes, M. Carlesso, and A. Bassi, Gravitational interaction through a feedback mechanism, Phys. Rev. D 103, 056011 (2021).
  14. J. Oppenheim, A postquantum theory of classical gravity? Phys. Rev. X 13, 041040 (2023).
  15. N. Piccione and A. Bassi, Hybrid classical-quantum Newtonian gravity with stable vacuum, Class. Quantum Grav. 42, 225002 (2025).
  16. S. Bose, A. Mazumdar, G. W. Morley, H. Ulbricht, M. Toroš, M. Paternostro, A. A. Geraci, P. F. Barker, M. S. Kim, and G. Milburn, Spin entanglement witness for quantum gravity, Phys. Rev. Lett. 119, 240401 (2017).
  17. C. Marletto and V. Vedral, Gravitationally induced entanglement between two massive particles is sufficient evidence of quantum effects in gravity, Phys. Rev. Lett. 119, 240402 (2017).
  18. L. Lami, J. S. Pedernales, and M. B. Plenio, Testing the quantumness of gravity without entanglement, Phys. Rev. X 14, 021022 (2024).
  19. O. Angeli, S. Donadi, G. D. Bartolomeo, J. L. Gaona-Reyes, A. Vinante, and A. Bassi, Probing the quantum nature of gravity through classical diffusion, arXiv:2501.13030.
  20. S. Bose, I. Fuentes, A. A. Geraci, S. M. Khan, S. Qvarfort, M. Rademacher, M. Rashid, M. Toroš, H. Ulbricht, and C. C. Wanjura, Massive quantum systems as interfaces of quantum mechanics and gravity, Rev. Mod. Phys. 97, 015003 (2025).
  21. S. Donadi, L. Ferialdi, and A. Bassi, Collapse dynamics are diffusive, Phys. Rev. Lett. 130, 230202 (2023).
  22. T. D. Galley, F. Giacomini, and J. H. Selby, Any consistent coupling between classical gravity and quantum matter is fundamentally irreversible, Quantum 7, 1142 (2023).
  23. A. Tilloy, General quantum-classical dynamics as measurement based feedback, SciPost Phys. 17, 083 (2024).
  24. A. Barchielli, Markovian dynamics for a quantum/classical system and quantum trajectories, J. Phys. A 57, 315301 (2024).
  25. D. Trillo and M. Navascués, Diósi-Penrose model of classical gravity predicts gravitationally induced entanglement, Phys. Rev. D 111, L121101 (2025).
  26. O. Angeli and M. Carlesso, Entanglement in Markovian hybrid classical-quantum theories of gravity, Phys. Rev. D 112, 024047 (2025).
  27. M. Toroš and A. Bassi, Bounds on quantum collapse models from matter-wave interferometry: Calculational details, J. Phys. A 51, 115302 (2018).
  28. N. Piccione and A. Bassi, Exploring the effects of mass dependence in spontaneous collapse models, Phys. Rev. A 112, 012212 (2025).
  29. L. Figurato, M. Dirindin, J. L. Gaona-Reyes, M. Carlesso, A. Bassi, and S. Donadi, On the effectiveness of the collapse in the Diósi–Penrose model, New J. Phys. 26, 113004 (2024).
  30. M. Carlesso, S. Donadi, L. Ferialdi, M. Paternostro, H. Ulbricht, and A. Bassi, Present status and future challenges of non-interferometric tests of collapse models, Nat. Phys. 18, 243 (2022).
  31. G. C. Ghirardi, A. Rimini, and T. Weber, Unified dynamics for microscopic and macroscopic systems, Phys. Rev. D 34, 470 (1986).
  32. P. Pearle, Combining stochastic dynamical state-vector reduction with spontaneous localization, Phys. Rev. A 39, 2277 (1989).
  33. G. C. Ghirardi, P. Pearle, and A. Rimini, Markov processes in Hilbert space and continuous spontaneous localization of systems of identical particles, Phys. Rev. A 42, 78 (1990).
  34. P. Pearle and E. Squires, Bound state excitation, nucleon decay experiments and models of wave function collapse, Phys. Rev. Lett. 73, 1 (1994).
  35. L. Diósi, A universal master equation for the gravitational violation of quantum mechanics, Phys. Lett. A 120, 377 (1987).
  36. L. Diósi, Models for universal reduction of macroscopic quantum fluctuations, Phys. Rev. A 40, 1165 (1989).
  37. R. Penrose, On gravity's role in quantum state reduction, Gen. Rel. Grav. 28, 581 (1996).
  38. R. Penrose, On the gravitization of quantum mechanics 1: Quantum state reduction, Found. Phys. 44, 557 (2014).
  39. S. Donadi, K. Piscicchia, C. Curceanu, L. Diósi, M. Laubenstein, and A. Bassi, Underground test of gravity-related wave function collapse, Nat. Phys. 17, 74 (2021).
  40. G. B. Folland and A. Sitaram, The uncertainty principle: A mathematical survey, J. Fourier Anal. Appl. 3, 207 (1997).
  41. S. McCurdy and R. Venkatraman, Quantitative stability for the Heisenberg–Pauli–Weyl inequality, Nonlin. Anal. 202, 112147 (2021).
  42. M. Fathi, A short proof of quantitative stability for the Heisenberg–Pauli–Weyl inequality, Nonlin. Anal. 210, 112403 (2021).
  43. V. V. Dodonov, Variance-based uncertainty relations: A concise review of inequalities discovered since 1927, Quantum Rep. 7, 34 (2025).
  44. N. Piccione, A proposal for a new kind of spontaneous collapse model, Found. Phys. 54, 4 (2024).
  45. N. Piccione, Gravitational Poissonian spontaneous localization model of hybrid quantum-classical Newtonian gravity: Energy increase and experimental bounds, arXiv:2602.19377.
  46. Q. Fu, Spontaneous radiation of free electrons in a nonrelativistic collapse model, Phys. Rev. A 56, 1806 (1997).
  47. S. Donadi, Electromagnetic radiation emission and flavour oscillations in collapse models, Ph.D. thesis, University of Trieste, 2014.
  48. S. Donadi and A. Bassi, The emission of electromagnetic radiation from a quantum system interacting with an external noise: A general result, J. Phys. A 48, 035305 (2015).
  49. M. Carlesso, A. Bassi, P. Falferi, and A. Vinante, Experimental bounds on collapse models from gravitational wave detectors, Phys. Rev. D 94, 124036 (2016).
  50. I. J. Arnquist, et al. (Majorana Collaboration), Search for spontaneous radiation from wave function collapse in the Majorana Demonstrator, Phys. Rev. Lett. 129, 080401 (2022).
  51. E. Aprile et al., Challenging spontaneous quantum collapse with XENONnT, Phys. Rev. Lett. 136, 120201 (2026).
  52. A. Tilloy and T. M. Stace, Neutron star heating constraints on wave-function collapse models, Phys. Rev. Lett. 123, 080402 (2019).
  53. S. L. Adler, A. Bassi, M. Carlesso, and A. Vinante, Testing continuous spontaneous localization with Fermi liquids, Phys. Rev. D 99, 103001 (2019).
  54. S. Donadi, K. Piscicchia, R. Del Grande, C. Curceanu, M. Laubenstein, and A. Bassi, Novel CSL bounds from the noise-induced radiation emission from atoms, Eur. Phys. J. C 81, 773 (2021).
  55. M. Janse, D. G. Uitenbroek, L. van Everdingen, J. Plugge, B. Hensen, and T. H. Oosterkamp, Current experimental upper bounds on spacetime diffusion, Phys. Rev. Res. 6, 033076 (2024).
  56. R. Bhattacharya and E. C. Waymire, A Basic Course in Probability Theory, 2nd ed. (Springer, 2016).
  57. L. C. Evans, Partial Differential Equations, 2nd ed., Graduate Studies in Mathematics (American Mathematical Society, 2010), Vol. 19.
  58. J. M. Borwein and J. D. Vanderwerff, Convex Functions: Constructions, Characterizations and Counterexamples (Cambridge University Press, 2010), Vol. 109.
  59. S. Kesavan, Symmetrization and Applications, Series in Analysis, Vol. 3 (World Scientific, 2006).
  60. E. H. Lieb and M. Loss, Analysis, 2nd ed., Graduate Studies in Mathematics, Vol. 14 (American Mathematical Society, 2001).
  61. H. M. Wiseman and G. J. Milburn, Quantum Measurement and Control, 1st ed. (Cambridge University Press, 2009).
  62. K. Jacobs, Quantum Measurement Theory and Its Applications (Cambridge University Press, 2014).
  63. S. L. Adler, Lower and upper bounds on CSL parameters from latent image formation and IGM heating, J. Phys. A 40, 2935 (2007).
  64. K. Piscicchia, S. Donadi, S. Manti, A. Bassi, M. Derakhshani, L. Diósi, and C. Curceanu, X-ray emission from atomic systems can distinguish between prevailing dynamical wave-function collapse models, Phys. Rev. Lett. 132, 250203 (2024).

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Phys. Rev. A 113, 062211– Published 9 June, 2026
  • Received 17 November 2025
  • Accepted 19 May 2026
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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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