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REVIEW 2 major objections 3 minor 54 references

Non-Markovian Poissonian Spontaneous Collapse Models

T0 review · 2 major / 3 minor · reviewed 2026-07-13 · grok-4.5

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Pith's one-line read A non-Markovian PSL model produces a long-time CPTP map identical to non-Markovian CSL, plus a time-local master equation and late-time flash statistics.

desk verdict Solid non-Markovian extension of PSL that recovers known maps and late-time flash intensities; the χ₀-neglect step is the only real soft spot and is openly heuristic. read the letter →

arxiv 2607.08955 v1 pith:TYSTLFJW submitted 2026-07-09 quant-ph

Nicolò Piccione This is my paper
classification quant-ph
keywords spontaneouscollapsenon-MarkoviandynamicsPoissonianlocalizationcontinuousCPTPmaptime-convolutionlessmasterequationsupercumulantexpansionflashprocess
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Spontaneous collapse models modify quantum evolution with stochastic terms so that macroscopic superpositions rapidly localize, addressing the measurement problem. This paper constructs a non-Markovian version of the Poissonian Spontaneous Localization model by letting each collapse point interact with matter over a finite time scale τ_C. In the appropriate continuum limit it derives an effective completely-positive trace-preserving map Φ_t that governs the average state of quantum matter for times much longer than τ_C, shows that this map can be made identical to the map of non-Markovian continuous spontaneous localization, obtains the associated time-convolutionless master equation via a supercumulant expansion, and gives the joint intensities of all late-time collapse flashes. The construction therefore supplies a well-defined flash-based non-Markovian collapse dynamics that recovers known Markovian limits and is presented as a natural stepping-stone toward relativistic extensions.

What carries the argument

The CPTP map Φ_t obtained after the Poissonian sprinkling average and the continuum (PSL) limit; it is written as a time-ordered exponential of a double integral involving the collapse operators and is the object that both matches non-Markovian CSL and generates the supercumulant expansion for the time-local master equation.

What would settle it

Compute the exact short-time flash intensity for a single particle prepared in a localized wave-packet with an explicitly correlated initial collapse-point state; if the late-time formula (55) continues to hold even for t_C ≲ τ_C, or if the difference fails to decay as t/τ_C o ∞, the central long-time claim is false.

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Extended reading notes

Core claim

In the PSL limit (interaction strength to zero, collapse-point density to infinity, product fixed at λ), the long-time statistical evolution of quantum matter starting from an initial state ρ_0 is given by the completely-positive trace-preserving map Φ_t of equation (23). That map can be chosen to coincide exactly with the map of non-Markovian continuous spontaneous localization models; the corresponding time-convolutionless master equation follows from the first non-vanishing supercumulant; and the joint intensities of all flashes occurring at times ≫ τ_C are given by the explicit formula (55).

Load-bearing premise

Any initial correlations between the quantum system and the collapse-point qubits become negligible for all observables and flash statistics once the evolution time is much larger than the interaction time scale τ_C.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 3 minor

Summary. The paper constructs a non-Markovian extension of the Poissonian Spontaneous Localization (PSL) model by allowing each Collapse Point (CP) to interact with quantum matter over a finite timescale τ_C. In the PSL continuum limit (Υ o0, μ_C o∞, Υ μ_C oλ), it derives an effective long-time (t≫ au_C) CPTP map Φ_t for the reduced dynamics of quantum matter starting from ρ_0 (Eq. 23), shows that this map can be specialized to recover the maps of non-Markovian CSL models, obtains the associated time-convolutionless master equation via a supercumulant expansion, and characterizes the multi-point flash intensity densities for late-time events (t_j≫ au_C) via Eq. (55).

Significance. If the long-time factorization assumption holds, the work supplies a mathematically consistent non-Markovian flash model that recovers both the Markovian PSL/CSL master equation and known non-Markovian Gaussian dynamics, while preserving a clear spacetime interpretation of flashes. The unitary unraveling, cell decomposition to white noise, identification with the Diósi–Ferialdi map, and the combinatorial extraction of flash intensities (Appendix B) are carefully executed technical contributions. The construction is a natural stepping-stone toward relativistic extensions that retain flashes, which is a recognized open direction in the field.

major comments (2)
  1. [Sec. III (Eqs. 13–15) and Sec. VI (Eq. 46)] Sections III (Eqs. 13–15) and VI (Eq. 46): The central claims that ρ_t=Φ_t ρ_0 for t≫ au_C and that the late-time flash intensities are given by Eq. (55) rest on neglecting the initial system–CP correlations χ_0. The supporting arguments (energy diffusion, spatial localization of excited CP-qubits near the support of ρ_0, and the vanishing relative weight of early flashes) are only heuristic. No operator-norm, trace-distance, or scaling estimate of the form ||Tr_CP{E[U(t)χ_0]}|| o0 (with t/ au_C or λ au_C^{3}/ au_S) is supplied. Because every subsequent formal result is conditional on this step, a controlled error bound or a sharper statement of the regime of validity is needed.
  2. [Sec. VI.B, Appendix B] Section VI.B and Appendix B: The derivation of the flash intensities correctly discards terms of order higher than Υ^{n/2} in the PSL limit and yields the compact expression (55). However, the same uncontrolled factorization (Eq. 46) is reused, so the intensities inherit the same limitation. The paper already notes that short-time intensities (t_C≲ au_C) become unphysical without χ_0; this observation should be elevated to a precise statement of the domain in which Eq. (55) is claimed to hold.
minor comments (3)
  1. [Sec. III–V] The super-operator notation (calligraphic fonts, the placeholder •) is introduced clearly in Sec. III, but a short reminder when the time-ordering T_L or T_C first appears would help readers less familiar with the supercumulant literature.
  2. [Sec. V, Eqs. (37)–(38)] The estimate (37)–(38) for the relative size of supercumulants is useful; stating the precise operator norm (or the physical scale that replaces it for unbounded operators) would make the truncation criterion fully self-contained.
  3. [Throughout] A few typographical concatenations appear in the arXiv source (e.g., “Spontaneouscollapsemodels”); these should be cleaned in the final version.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the long-time map Φ_t, TCL equation and late-time flash intensities are derived from the defined CP-qubit model plus the PSL continuum limit; self-citations only supply the Markovian starting point.

full rationale

The paper defines a non-Markovian PSL model via Poisson sprinkling of CP-qubits that interact for a finite time τ_C, then takes the explicit continuum limit Υ o0, μ_C o∞, Υ μ_C oλ. From the resulting Gaussian unraveling it obtains the CPTP map Φ_t (Eq. 23) by direct identification with the known Gaussian channel of Diósi–Ferialdi, recovers the Adler–Bassi non-Markovian CSL map under a linear filter, extracts the second-order supercumulant TCL generator, and constructs the late-time flash intensities μ_F^(n) by the same limit applied to the projective measurement probabilities. All steps are algebraic or combinatorial consequences of the model definition and the stated limit; no free parameters are fitted to data, no uniqueness theorem is imported from the author’s prior work to force the result, and no quantity is redefined as its own prediction. The only self-citations ([19,33,36,37]) introduce the original Markovian PSL construction that is being extended; they are not used to justify the non-Markovian map or the flash formulae. The heuristic neglect of initial correlations χ_0 is an uncontrolled approximation, not a circular step. Hence the derivation chain is self-contained against its own inputs.

Assumptions & free parameters 3 free parameters · 5 assumptions · 2 invented entities

The central claims rest on the postulated existence of Poisson-distributed Collapse Points carrying qubits, the PSL continuum limit, the finite interaction time τ_C, and the heuristic that initial correlations χ_0 become irrelevant at late times. No empirical free parameters are fitted; λ, τ_C and the spatial smearing are free model parameters inherited from the collapse-model program.

free parameters (3)
  • λ (collapse rate density)
    Product Υ μ_C kept finite in the PSL limit; sets the overall strength of the noise and appears as a free scale in Φ_t and the flash intensities.
  • τ_C (interaction/correlation time)
    Characteristic duration of each CP-matter interaction; introduced by hand to generate non-Markovianity and controls the validity of the long-time approximation.
  • r_C (spatial smearing length)
    Width of the mass-density smearing function in the mass-proportional version; free phenomenological parameter carried over from Markovian PSL.
assumptions (5)
  • ad hoc to paper Spacetime is filled with a Poisson sprinkling of Collapse Points of constant density μ_C, each carrying a qubit initialized in |0⟩.
    Foundational postulate of the PSL framework (Sec. II); not derived from more primitive principles.
  • ad hoc to paper Each CP interacts via √Υ L̂_xC(t) ⊗ X̂ and is projectively measured in the Z basis at its time coordinate, producing a flash if the outcome is |1⟩.
    Defines the microscopic dynamics of the model (Sec. II).
  • domain assumption The PSL continuum limit Υ o0, μ_C o∞ with Υ μ_C oλ>0 yields a well-defined Gaussian white-noise unraveling.
    Standard continuum limit used throughout the collapse-model literature; invoked in Sec. IV.
  • ad hoc to paper Initial correlations χ_0 between matter and future CP-qubits become negligible for t≫ au_C, so that the reduced dynamics is given by the factorized map Φ_t.
    Heuristic assumption stated at the end of Sec. III and reused in Sec. VI; no rigorous bound is supplied.
  • standard math Supercumulant expansion of a Gaussian map truncates at second order when λ au_C^{3}/ au_S ≪1.
    Taken from the open-systems literature (Refs. [39,40]); used in Sec. V.
invented entities (2)
  • Collapse Points (CPs) with associated CP-qubits
    purpose: Provide a spacetime-localized microscopic mechanism that generates both the stochastic collapse dynamics and the discrete flash process.
    Postulated objects of the PSL model; no independent experimental evidence is claimed beyond the phenomenological success of collapse models in general.
  • Finite interaction time τ_C for each CP
    purpose: Generate non-Markovian correlations while recovering Markovian PSL as τ_C o0.
    New parameter introduced in this paper; its value is free and not fixed by independent data.

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Cite this review

Pith. "Pith review of Non-Markovian Poissonian Spontaneous Collapse Models." pith.science (2026). https://pith.science/paper/TYSTLFJW

@misc{pith2026260708955,
  author       = {Pith},
  title        = {Pith review of: Non-Markovian Poissonian Spontaneous Collapse Models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TYSTLFJW}},
  note         = {Machine review of arXiv:2607.08955}
}
abstract

Spontaneous collapse models provide a possible solution to the measurement problem by modifying standard quantum dynamics. The modification consists of adding non-linear and stochastic terms inducing wavefunction collapse in space. Non-Markovian versions of these models are motivated by physical reasons, phenomenological consistency, and potential for relativistic extensions. Here, we investigate a non-Markovian version of the Poissonian Spontaneous Localization (PSL) model, i.e., a model characterized by instantaneous and localized collapse events. We assume that our model is characterized by a typical time scale $\tau_C$ so that, given an initial state $\rho_0$ of standard quantum matter at time $t=0$, we derive an effective long-time ($t\gg \tau_C$) statistical dynamics in terms of a CPTP map $\Phi_t$. We then show how $\Phi_t$ can be made equal to that obtained by non-Markovian CSL models. Moreover, given $\Phi_t$, we obtain the associated time-convolutionless master equation by means of a supercumulant expansion. Finally, we characterize the collapse events process (for events with $t \gg \tau_C$) given an initial quantum state $\rho_0$ at $t=0$.

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Works this paper leans on

54 extracted references · 6 linked inside Pith

  1. [1]

    S.Bell,Speakable and unspeakable in quantum mechan- ics: Collected papers on quantum philosophy(Cambridge university press, 2004)

    J. S.Bell,Speakable and unspeakable in quantum mechan- ics: Collected papers on quantum philosophy(Cambridge university press, 2004)

    2004

  2. [2]

    Norsen,Foundations of quantum mechanics(Springer, 2017)

    T. Norsen,Foundations of quantum mechanics(Springer, 2017)

    2017

  3. [3]

    Dürr and D

    D. Dürr and D. Lazarovici,Understanding quantum me- chanics(Springer, 2020)

    2020

  4. [4]

    Tumulka,Foundations of quantum mechanics, Vol

    R. Tumulka,Foundations of quantum mechanics, Vol. 1003 (Springer Nature, 2022)

    2022

  5. [5]

    Bassi and G

    A. Bassi and G. Ghirardi,Dynamical reduction models, Physics Reports379, 257–426 (2003)

    2003

  6. [6]

    Bassi, K

    A. Bassi, K. Lochan, S. Satin, T. P. Singh, and H. Ul- bricht,Models of wave-function collapse, underlying the- ories, and experimental tests, Rev. Mod. Phys.85, 471– 527 (2013)

    2013

  7. [7]

    Bassi, M

    A. Bassi, M. Dorato, and H. Ulbricht,Collapse models: A theoretical, experimental and philosophical review, En- tropy25, 645 (2023)

    2023

  8. [8]

    G. C. Ghirardi, A. Rimini, and T. Weber,Unified dy- namics for microscopic and macroscopic systems, Phys. Rev. D34, 470–491 (1986)

    1986

Show all 54 references
  1. [9]

    L.Diósi,A universal master equation for the gravitational violation of quantum mechanics, Physics Letters A120, 377–381 (1987)

    1987

  2. [10]

    Diósi,Models for universal reduction of macroscopic quantum fluctuations, Phys

    L. Diósi,Models for universal reduction of macroscopic quantum fluctuations, Phys. Rev. A40, 1165–1174 (1989)

    1989

  3. [11]

    Pearle,Combining stochastic dynamical state-vector reduction with spontaneous localization, Phys

    P. Pearle,Combining stochastic dynamical state-vector reduction with spontaneous localization, Phys. Rev. A39, 2277–2289 (1989)

    1989

  4. [12]

    G. C. Ghirardi, P. Pearle, and A. Rimini,Markov pro- cesses in hilbert space and continuous spontaneous local- ization of systems of identical particles, Phys. Rev. A42, 78–89 (1990)

    1990

  5. [13]

    S. L. Adler and A. Bassi,Collapse models with non-white noises, Journal of Physics A: Mathematical and Theoret- ical40, 15083–15098 (2007), arXiv:0708.3624 [quant-ph]

    2007 arXiv

  6. [14]

    S. L. Adler and A. Bassi,Collapse models with non- white noises II: particle-density coupled noises, Journal of Physics A: Mathematical and Theoretical41, 395308 (2008), arXiv:0807.2846 [quant-ph]

    2008 arXiv

  7. [15]

    Donadi,Electromagnetic Radiation Emission and Flavour Oscillations in Collapse Models, Ph.D

    S. Donadi,Electromagnetic Radiation Emission and Flavour Oscillations in Collapse Models, Ph.D. thesis, University of Trieste (2014)

    2014

  8. [16]

    Donadi and A

    S. Donadi and A. Bassi,The emission of electromag- netic radiation from a quantum system interacting with an external noise: a general result, Journal of Physics A: Mathematical and Theoretical48, 035305 (2015)

    2015

  9. [17]

    Donadi, K

    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, The Euro- pean Physical Journal C81, 773 (2021)

    2021

  10. [18]

    Donadi, K

    S. Donadi, K. Piscicchia, C. Curceanu, L. Diósi, M. Laubenstein, and A. Bassi,Underground test of gravity-related wave function collapse,NaturePhysics17, 74–78 (2021)

    2021

  11. [19]

    Piccione and A

    N. Piccione and A. Bassi,Exploring the effects of mass dependence in spontaneous collapse models, Phys. Rev. A 112, 012212 (2025)

    2025

  12. [20]

    W. C. Myrvold,Relativistic markovian dynamical col- lapse theories must employ nonstandard degrees of free- dom, Phys. Rev. A96, 062116 (2017)

    2017

  13. [21]

    Jones, G

    C. Jones, G. Gasbarri, and A. Bassi,Mass-coupled rel- ativistic spontaneous collapse models, J. Phys. A54, 295306 (2021)

    2021

  14. [22]

    Diósi,Is there a relativistic gorini-kossakowski- lindblad-sudarshan master equation?, Phys

    L. Diósi,Is there a relativistic gorini-kossakowski- lindblad-sudarshan master equation?, Phys. Rev. D106, L051901 (2022)

    2022

  15. [23]

    Gundhi, L

    A. Gundhi, L. Diósi, and M. Carlesso, Towards relativistic generalization of collapse models (2025), arXiv:2507.06954 [quant-ph]

    2025 arXiv

  16. [24]

    Tumulka,A relativistic version of the ghirardi–rimini– weber model, J

    R. Tumulka,A relativistic version of the ghirardi–rimini– weber model, J. Stat. Phys125, 821–840 (2006)

    2006

  17. [25]

    R.Tumulka,The point processes of the grw theory of wave function collapse, Reviews in Mathematical Physics21, 155–227 (2009)

    2009

  18. [26]

    D. J. Bedingham,Relativistic state reduction dynamics, Found. Phys41, 686–704 (2011)

    2011

  19. [27]

    Bassi and L

    A. Bassi and L. Ferialdi,Non-Markovian quantum tra- jectories: An exact result, Physical Review Letters103, 050403 (2009), arXiv:0907.1615 [quant-ph]

    2009 arXiv

  20. [28]

    Diósi and L

    L. Diósi and L. Ferialdi,General non-markovian struc- ture of gaussian master and stochastic schrödinger equa- tions, Phys. Rev. Lett.113, 200403 (2014)

    2014

  21. [29]

    Tilloy and H

    A. Tilloy and H. M. Wiseman,Non-Markovian wave- function collapse models are Bohmian-like theories in dis- guise, Quantum5, 594 (2021)

    2021

  22. [30]

    Carlesso, L

    M. Carlesso, L. Ferialdi, and A. Bassi,Colored collapse models from the non-interferometric perspective, The Eu- ropean Physical Journal D72, 159 (2018)

    2018

  23. [31]

    Toroš, G

    M. Toroš, G. Gasbarri, and A. Bassi,Colored and dis- sipative continuous spontaneous localization model and bounds from matter-wave interferometry, Physics Letters A381, 3921–3927 (2017), arXiv:1601.03672 [quant-ph]

    2017 arXiv

  24. [32]

    M.TorošandA.Bassi,Bounds on quantum collapse mod- els from matter-wave interferometry: calculational de- tails,JournalofPhysicsA:MathematicalandTheoretical 51, 115302 (2018)

    2018

  25. [33]

    Piccione,A proposal for a new kind of spontaneous collapse model, Found

    N. Piccione,A proposal for a new kind of spontaneous collapse model, Found. Phys.54, 4 (2024)

    2024

  26. [34]

    Ciccarello, S

    F. Ciccarello, S. Lorenzo, V. Giovannetti, and G. M. Palma,Quantum collision models: open system dynam- ics from repeated interactions, Physics Reports954, 1–70 (2022)

    2022

  27. [35]

    Tumulka,On spontaneous wave function collapse and quantum field theory, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences462, 1897–1908 (2006)

    R. Tumulka,On spontaneous wave function collapse and quantum field theory, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences462, 1897–1908 (2006)

    1908

  28. [36]

    Piccione and A

    N. Piccione and A. Bassi,Hybrid classical-quantum new- tonian gravity with stable vacuum, Classical and Quan- tum Gravity42, 225002 (2025)

    2025

  29. [37]

    N. Piccione, Gravitational poissonian spontaneous lo- 10 calization model of hybrid quantum-classical newto- nian gravity: Energy increase and experimental bounds (2026), arXiv:2602.19377 [quant-ph]

    2026

  30. [38]

    Breuer and F

    H.-P. Breuer and F. Petruccione,The Theory of Open quantum systems(Oxford University Press, New York, 2007)

    2007

  31. [39]

    Szańkowski,Introduction to the theory of open quan- tum systems, SciPost Phys

    P. Szańkowski,Introduction to the theory of open quan- tum systems, SciPost Phys. Lect. Notes , 68 (2023)

    2023

  32. [40]

    Szańkowski,Introduction to the theory of open quan- tum systems, Submission (2025)

    P. Szańkowski,Introduction to the theory of open quan- tum systems, Submission (2025)

    2025

  33. [41]

    Surya,The causal set approach to quantum grav- ity, Living Reviews in Relativity22, 5 (2019), arXiv:1903.11544 [gr-qc]

    S. Surya,The causal set approach to quantum grav- ity, Living Reviews in Relativity22, 5 (2019), arXiv:1903.11544 [gr-qc]

    2019 arXiv

  34. [42]

    D. J. Bedingham, inDo Wave Functions Jump? (Springer, 2021) pp. 191–203

    2021

  35. [43]

    Bedingham, S

    D. Bedingham, S. K. Modak, and D. Sudarsky,Relativis- tic collapse dynamics and black hole information loss, Phys. Rev. D94, 045009 (2016)

    2016

  36. [44]

    B. A. Juárez-Aubry, B. S. Kay, and D. Sudarsky,Gen- erally covariant dynamical reduction models and the hadamard condition, Phys. Rev. D97, 025010 (2018)

    2018

  37. [45]

    Donadi, L

    S. Donadi, L. Ferialdi, and A. Bassi,Collapse dynamics are diffusive, Phys. Rev. Lett.130, 230202 (2023)

    2023

  38. [46]

    Øksendal,Stochastic Differential Equations: An In- troduction with Applications, 6th ed

    B. Øksendal,Stochastic Differential Equations: An In- troduction with Applications, 6th ed. (Springer, Berlin, 2003)

    2003

  39. [47]

    Schondorf, The wiener measure and donsker’s invari- ance principle (2019), expository manuscript

    E. Schondorf, The wiener measure and donsker’s invari- ance principle (2019), expository manuscript

    2019

  40. [48]

    Durrett,Probability: Theory and Examples, 5th ed

    R. Durrett,Probability: Theory and Examples, 5th ed. (Cambridge University Press, 2019)

    2019

  41. [49]

    S. L. Adler,Lower and upper bounds on csl parameters from latent image formation and igm heating, Journal of Physics A: Mathematical and Theoretical40, 2935 (2007)

    2007

  42. [50]

    Figurato, M

    L. Figurato, M. Dirindin, J. L. Gaona-Reyes, M. Car- lesso, A. Bassi, and S. Donadi,On the effectiveness of the collapse in the diósi–penrose model, New Journal of Physics26, 113004 (2024)

    2024

  43. [51]

    Pechukas,Reduced dynamics need not be completely positive, Physical Review Letters73, 1060–1062 (1994)

    P. Pechukas,Reduced dynamics need not be completely positive, Physical Review Letters73, 1060–1062 (1994)

    1994

  44. [52]

    de Vega and D

    I. de Vega and D. Alonso,Dynamics of non-markovian open quantum systems, Reviews of Modern Physics89, 015001 (2017)

    2017

  45. [53]

    D. J. Daley and D. Vere-Jones,An Introduction to the Theory of Point Processes. Volume I: Elementary Theory and Methods, 2nd ed., Probability and Its Applications (Springer, New York, NY, 2003)

    2003

  46. [54]

    D. J. Daley and D. Vere-Jones,An Introduction to the Theory of Point Processes. Volume II: General Theory and Structure, 2nd ed., Probability and Its Applications (Springer, New York, NY, 2008). 11 Appendix A: Flash intensities completely characterize the stochastic flash proc...

    2008

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