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  • Open Access

Quantifying multitime memory in sequential measurement statistics through violations of the quantum regression theorem

Paolo Luppi*, Claudia Benedetti†, and Andrea Smirne‡

  • *Contact author: paolo.luppi@unimi.it
  • †Contact author: claudia.benedetti@unimi.it
  • ‡Contact author: andrea.smirne@unimi.it
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Phys. Rev. A 114, 052204 – Published 4 November, 2026

DOI: https://doi.org/10.1103/ssf1-j2j8

Abstract

We investigate the presence of memory in the sequential measurement statistics of an open quantum system, as witnessed by the departure from the quantum regression theorem (QRT), that is, the possibility to predict multitime probabilities from one-time reduced dynamical maps. For factorized initial states, we identify an exact decomposition of the two-time propagator into a QRT-like contribution, fully determined by the reduced dynamical map, and a memory term encoding system-environment correlations across the intervention. In the weak-coupling regime, the memory term yields an explicit second-order correction expressed in terms of the reduced map and bath correlation functions. Furthermore, we introduce an operational quantifier of QRT violations based on the distance between exact and QRT-predicted joint probabilities. Benchmarking the framework on a spin-boson model and using a pseudomode embedding as nonperturbative reference, we comprehensively analyze the impact of spectral-density parameters, environmental temperature, and measurement protocols on the non-Markovianity of the multitime statistics. Comparison with a one-time witness shows that reduced-state non-Markovianity and multitime memory are related but inequivalent: the latter, as probed through sequential statistics, is intrinsically protocol dependent and can become visible at higher temporal orders even when lower-order statistics remain compatible with QRT predictions.

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References (63)

  1. H. J. Carmichael, An Open Systems Approach to Quantum Optics, Lecture Notes in Physics Monographs Vol. 18 (Springer, Berlin, 1993).
  2. S. Mukamel, Principles of Nonlinear Optical Spectroscopy (Oxford University Press, New York, 1995).
  3. H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, Oxford, 2002).
  4. C. W. Gardiner and P. Zoller, Quantum Noise: A Handbook of Markovian and Non-Markovian Quantum Stochastic Methods with Applications toQuantum Optics, 3rd ed., Springer Series in Synergetics (Springer, Berlin, 2004).
  5. A. Rivas and S. F. Huelga, Open Quantum Systems: An Introduction, SpringerBriefs in Physics (Springer, Berlin, 2011).
  6. B. Vacchini, Open Quantum Systems (Springer, Cham, Switzerland, 2024).
  7. M. Lax, Formal theory of quantum fluctuations from a driven state, Phys. Rev. 129, 2342 (1963).
  8. M. Lax, Quantum noise. XI. Multitime correspondence between quantum and classical stochastic processes, Phys. Rev. 172, 350 (1968).
  9. R. Dümcke, Convergence of multitime correlation functions in the weak and singular coupling limits, J. Math. Phys. 24, 311 (1983).
  10. S. Swain, Master equation derivation of quantum regression theorem, J. Phys. A: Math. Gen. 14, 2577 (1981).
  11. D. Alonso and I. de Vega, Multiple-time correlation functions for non-Markovian interaction: Beyond the quantum regression theorem, Phys. Rev. Lett. 94, 200403 (2005).
  12. A. Ivanov and H.-P. Breuer, Extension of the Nakajima-Zwanzig approach to multitime correlation functions of open systems, Phys. Rev. A 92, 032113 (2015).
  13. D. P. S. McCutcheon, Optical signatures of non-Markovian behavior in open quantum systems, Phys. Rev. A 93, 022119 (2016).
  14. M. Ban, S. Kitajima, and F. Shibata, Two-time correlation function of an open quantum system in contact with a Gaussian reservoir, Phys. Rev. A 97, 052101 (2018).
  15. M. Bonifacio and A. A. Budini, Perturbation theory for operational quantum non-Markovianity, Phys. Rev. A 102, 022216 (2020).
  16. D. Lonigro and D. Chruściński, Quantum regression in dephasing phenomena, J. Phys. A: Math. Theor. 55, 225308 (2022).
  17. M. Salamon, O. Dudgeon, A. Nazir, and J. Iles-Smith, Markovian approach to n-photon correlations beyond the quantum regression theorem, Phys. Rev. Lett. 136, 080401 (2026).
  18. G. Lindblad, Non-Markovian quantum stochastic processes and their entropy, Commun. Math. Phys. 65, 281 (1979).
  19. L. Accardi, A. Frigerio, and J. T. Lewis, Quantum stochastic processes, Publ. Res. Inst. Math. Sci. 18, 97 (1982).
  20. K. Modi, Operational approach to open dynamics and quantifying initial correlations, Sci. Rep. 2, 581 (2012).
  21. F. A. Pollock, C. Rodríguez-Rosario, T. Frauenheim, M. Paternostro, and K. Modi, Operational Markov condition for quantum processes, Phys. Rev. Lett. 120, 040405 (2018).
  22. P. Taranto, F. A. Pollock, S. Milz, M. Tomamichel, and K. Modi, Quantum Markov order, Phys. Rev. Lett. 122, 140401 (2019).
  23. S. Milz and K. Modi, Quantum stochastic processes and quantum non-Markovian phenomena, PRX Quantum 2, 030201 (2021).
  24. C. Giarmatzi and F. Costa, Witnessing quantum memory in non-Markovian processes, Quantum 5, 440 (2021).
  25. B. Vacchini, A. Smirne, E.-M. Laine, J. Piilo, and H.-P. Breuer, Markovianity and non-Markovianity in quantum and classical systems, New J. Phys. 13, 093004 (2011).
  26. G. Guarnieri, A. Smirne, and B. Vacchini, Quantum regression theorem and non-Markovianity of quantum dynamics, Phys. Rev. A 90, 022110 (2014).
  27. S. Milz, M. S. Kim, F. A. Pollock, and K. Modi, Completely positive divisibility does not mean Markovianity, Phys. Rev. Lett. 123, 040401 (2019).
  28. L. Li, M. J. W. Hall, and H. M. Wiseman, Concepts of quantum non-Markovianity: A hierarchy, Phys. Rep. 759, 1 (2018).
  29. M. M. Wolf, J. Eisert, T. S. Cubitt, and J. I. Cirac, Assessing non-Markovian quantum dynamics, Phys. Rev. Lett. 101, 150402 (2008).
  30. Á. Rivas, S. F. Huelga, and M. B. Plenio, Entanglement and non-Markovianity of quantum evolutions, Phys. Rev. Lett. 105, 050403 (2010).
  31. S. Wißmann, H.-P. Breuer, and B. Vacchini, Generalized trace-distance measure connecting quantum and classical non-Markovianity, Phys. Rev. A 92, 042108 (2015).
  32. H.-P. Breuer, E.-M. Laine, J. Piilo, and B. Vacchini, Colloquium: Non-Markovian dynamics in open quantum systems, Rev. Mod. Phys. 88, 021002 (2016).
  33. D. Chruściński, Dynamical maps beyond Markovian regime, Phys. Rep. 992, 1 (2022).
  34. V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, Completely positive dynamical semigroups of n‐level systems, J. Math. Phys. 17, 821 (1976).
  35. G. Lindblad, On the generators of quantum dynamical semigroups, Commun. Math. Phys. 48, 119 (1976).
  36. E. B. Davies, Markovian master equations, Commun. Math. Phys. 39, 91 (1974).
  37. C. Bäcker, V. Link, and W. T. Strunz, Verifying quantum memory in the dynamics of spin-boson models, Phys. Rev. Res. 8, 023106 (2026).
  38. L. S. V. Santos, O. Gühne, and S. Nimmrichter, Two-point measurement correlations beyond the quantum regression theorem, arXiv:2507.06088.
  39. B. M. Garraway, Nonperturbative decay of an atomic system in a cavity, Phys. Rev. A 55, 2290 (1997).
  40. D. Tamascelli, A. Smirne, S. F. Huelga, and M. B. Plenio, Nonperturbative treatment of non-Markovian dynamics of open quantum systems, Phys. Rev. Lett. 120, 030402 (2018).
  41. D. Chruściński, A. Kossakowski, and Á. Rivas, Measures of non-Markovianity: Divisibility versus backflow of information, Phys. Rev. A 83, 052128 (2011).
  42. R. Kubo, The fluctuation-dissipation theorem, Rep. Prog. Phys. 29, 255 (1966).
  43. A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, Introduction to quantum noise, measurement, and amplification, Rev. Mod. Phys. 82, 1155 (2010).
  44. T. Heinosaari and M. Ziman, The Mathematical Language of Quantum Theory: From Uncertainty to Entanglement (Cambridge University Press, Cambridge, Enlgand, 2011).
  45. A. Smirne, D. Egloff, M. G. Díaz, M. B. Plenio, and S. F. Huelga, Coherence and non-classicality of quantum Markov processes, Quantum Sci. Technol. 4, 01LT01 (2019).
  46. S. Milz, D. Egloff, P. Taranto, T. Theurer, M. B. Plenio, A. Smirne, and S. F. Huelga, When is a non-Markovian quantum process classical? Phys. Rev. X 10, 041049 (2020).
  47. A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A. Fisher, A. Garg, and W. Zwerger, Dynamics of the dissipative two-state system, Rev. Mod. Phys. 59, 1 (1987).
  48. H.-P. Breuer, B. Kappler, and F. Petruccione, Stochastic wave-function method for non-Markovian quantum master equations, Phys. Rev. A 59, 1633 (1999).
  49. B. Vacchini and H.-P. Breuer, Exact master equations for the non-Markovian decay of a qubit, Phys. Rev. A 81, 042103 (2010).
  50. F. Albarelli, B. Vacchini, and A. Smirne, Pseudomode treatment of strong-coupling quantum thermodynamics, Quantum Sci. Technol. 10, 015041 (2025).
  51. F. Chen, E. Arrigoni, and M. Galperin, Markovian treatment of non-Markovian dynamics of open fermionic systems, New J. Phys. 21, 123035 (2019).
  52. A. Smirne, D. Tamascelli, J. Lim, M. B. Plenio, and S. F. Huelga, Non-perturbative treatment of open-system multi-time expectation values in Gaussian bosonic environments, Open Syst. Inf. Dyn. 29, 2250019 (2022).
  53. J. R. Johansson, P. D. Nation, and F. Nori, QuTiP: An open-source Python framework for the dynamics of open quantum systems, Comput. Phys. Commun. 183, 1760 (2012).
  54. J. R. Johansson, P. D. Nation, and F. Nori, QuTiP 2: A Python framework for the dynamics of open quantum systems, Comput. Phys. Commun. 184, 1234 (2013).
  55. M. Beth Ruskai, S. Szarek, and E. Werner, An analysis of completely-positive trace-preserving maps on M2, Linear Algebra Appl. 347, 159 (2002).
  56. R. Swinbank and R. James Purser, Fibonacci grids: A novel approach to global modelling, Q. J. R. Meteorol. Soc. 132, 1769 (2006).
  57. Á. González, Measurement of areas on a sphere using Fibonacci and latitude-longitude lattices, Math. Geosci. 42, 49 (2010).
  58. D. Braak, Integrability of the Rabi model, Phys. Rev. Lett. 107, 100401 (2011).
  59. L. Bernazzani, B. Gulácsi, and G. Burkard, Universal dissipators for driven open quantum systems and the correction to linear response, Quantum Sci. Technol. 10, 045050 (2025).
  60. S. Milz, F. A. Pollock, and K. Modi, An introduction to operational quantum dynamics, Open Syst. Inf. Dyn. 24, 1740016 (2017).
  61. P. Strasberg and M. G. Díaz, Classical quantum stochastic processes, Phys. Rev. A 100, 022120 (2019).
  62. P. Strasberg, Classicality with(out) decoherence: Concepts, relation to Markovianity, and a random matrix theory approach, SciPost Phys. 15, 024 (2023).
  63. P. Luppi, C. Benedetti, and A. Smirne, Temporal nonclassicality in continuous-time quantum walks, Phys. Rev. A 113, 042212 (2026).

Outline

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Phys. Rev. A 114, 052204– Published 4 November, 2026
  • Received 7 May 2026
  • Accepted 20 October 2026
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