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arxiv: 2606.03848 · v1 · pith:BPKT6YL4new · submitted 2026-06-02 · 🪐 quant-ph · cond-mat.stat-mech

Generating quantum ensembles via reverse-time quantum diffusions

Ma\"el Bompais , M\u{a}d\u{a}lin Gu\c{t}\u{a} , Juan P. Garrahan This is my paper

Pith reviewed 2026-06-28 09:40 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.stat-mech
keywords quantum diffusionreverse time dynamicsstochastic Schrödinger equationquantum trajectoriesgenerative modelsdenoisingcontinuous measurementquantum ensembles
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The pith

Reverse-time dynamics for quantum diffusions allows generating complex quantum ensembles from simple distributions.

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

This paper develops a theory for reversing the time evolution of quantum systems undergoing continuous measurement and diffusion. It shows that the reversed process can denoise quantum trajectories by starting from an easy-to-sample distribution and reaching a target ensemble of quantum states. The key is that the reverse dynamics is itself a valid physical process, using the original noise but with an added feedback term dependent on the current state. This mirrors how classical diffusion models use a score function to generate data. The framework includes ways to learn the reverse process from observed forward trajectories and to start the denoising using purification techniques.

Core claim

The authors derive the exact reverse-time dynamics for quantum trajectories from the forward stochastic Schrödinger equation, proving that it coincides with the time-reversal of the original process. This denoising dynamics is shown to be a physically admissible quantum diffusion featuring the same measurement-induced noise accompanied by a state-dependent feedback Hamiltonian, which serves as the direct quantum analogue of the score function in classical generative diffusion models. Consequently, samples from a simple initial distribution can be transformed into samples from a more complex ensemble of quantum states through this learned reverse process.

What carries the argument

The state-dependent feedback Hamiltonian in the reverse stochastic Schrödinger equation, functioning as the quantum version of the classical score function to steer the denoising.

If this is right

  • Samples from simple distributions can be converted into those of complex quantum ensembles via the denoising dynamics.
  • The denoising dynamics can be learnt directly from forward trajectory data.
  • Purification techniques can initialise the denoising process.
  • The reverse process remains physically admissible with the original measurement noise.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This method could be applied to simulate or generate states in quantum many-body systems where direct sampling is hard.
  • Efficient approximation of the feedback term might lead to scalable quantum generative algorithms.
  • The framework suggests connections between continuous measurement theory and quantum machine learning for state preparation.

Load-bearing premise

An exact time-reversal of the forward noising stochastic Schrödinger equation exists and yields a physically valid quantum diffusion process.

What would settle it

Running the derived reverse dynamics on simulated forward trajectories of a two-level quantum system and checking whether the statistics of the generated states match those expected from time-reversed measurements.

Figures

Figures reproduced from arXiv: 2606.03848 by Juan P. Garrahan, Ma\"el Bompais, M\u{a}d\u{a}lin Gu\c{t}\u{a}.

Figure 1
Figure 1. Figure 1: FIG. 1: (a) Forward (noising) dynamics for the 2-level [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Norm of the average reverse Hamiltonian in [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
read the original abstract

We establish a reverse-time denoising theory for quantum diffusions of continuously measured quantum systems. Starting from the stochastic Schr\"odinger equation of a forward noising dynamics, we derive the exact reverse-time dynamics for quantum trajectories, whose law coincides with the time-reversal of the original process. We prove that the denoising dynamics is a physically admissible quantum diffusion, with the same measurement-induced noise but a state-dependent feedback Hamiltonian, a direct analogue of the "score function" of generative classical diffusion models. This provides a principled framework for converting samples of a simple distribution into those of a more complex ensemble of quantum states. We show how the denoising dynamics can be directly learnt from forward trajectory data, and how to exploit purification to initialise the denoising process.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper establishes a reverse-time denoising theory for quantum diffusions arising from continuously measured quantum systems. Starting from the forward stochastic Schrödinger equation, it derives the exact reverse-time dynamics whose law matches the time-reversal of the original process. It proves that this denoising dynamics is a physically admissible quantum diffusion (same measurement-induced noise, but with a state-dependent feedback Hamiltonian that plays the role of the classical score function). The work also addresses learning the reverse dynamics from forward trajectory data and using purification to initialize the denoising process, with the goal of converting samples from a simple distribution into samples from a more complex ensemble of quantum states.

Significance. If the central derivation and admissibility proof hold, the manuscript supplies a principled quantum counterpart to score-based generative diffusion models. This could enable new methods for sampling complex quantum state distributions without ad-hoc fitting, with potential applications in quantum state preparation and ensemble generation. The explicit time-reversal construction and the data-driven learning component are notable strengths.

minor comments (3)
  1. The abstract and introduction would benefit from a brief statement of the precise form of the forward stochastic Schrödinger equation (including the measurement operators and noise terms) to make the starting point fully explicit for readers.
  2. Notation for the state-dependent feedback Hamiltonian and the reverse-time process should be introduced with a clear comparison table or side-by-side equations against the classical score function to highlight the analogy.
  3. Section on learning the dynamics from data should specify the loss function or objective used for training and any regularization terms that ensure physical admissibility is preserved during learning.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. The referee's description accurately captures the derivation of the reverse-time dynamics, the admissibility proof, the learning procedure, and the use of purification. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

Derivation self-contained from forward stochastic Schrödinger equation

full rationale

The paper derives the reverse-time quantum diffusion directly from the forward stochastic Schrödinger equation of a continuously measured system, proving admissibility with the same noise and a state-dependent feedback term. No quoted steps reduce by definition, fit, or self-citation chain to the target result; the time-reversal construction is a standard mathematical operation on the given forward process and does not rely on fitted parameters renamed as predictions or uniqueness theorems imported from the authors' prior work. The analogy to classical score functions follows as a consequence rather than an assumption. This is the most common honest non-finding for a derivation paper.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on standard quantum trajectory theory; the main addition is the reverse dynamics construction. No free parameters are mentioned. One domain assumption and one invented entity are identified.

axioms (1)
  • domain assumption Forward dynamics governed by the stochastic Schrödinger equation for continuously measured quantum systems.
    Explicit starting point stated in the abstract for deriving the reverse process.
invented entities (1)
  • state-dependent feedback Hamiltonian no independent evidence
    purpose: To realize the reverse dynamics as a physically admissible quantum diffusion analogous to the classical score function.
    Introduced in the abstract as the key component of the denoising dynamics; no independent evidence outside the derivation is provided.

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discussion (0)

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Reference graph

Works this paper leans on

38 extracted references · 12 canonical work pages · 7 internal anchors

  1. [1]

    Generating quantum ensembles via reverse-time quantum diffusions

    for details. At long times the initialµ 0(ψ) is reconstructed, thus transforming noisy initial pure states to samples of the distribution of interest. terms of genuine quantum stochastic trajectories. Here we present a general framework that addresses all of these issues. We develop a reverse-time construction for quantum arXiv:2606.03848v1 [quant-ph] 2 J...

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  2. [2]

    Bond-Taylor, A

    S. Bond-Taylor, A. Leach, Y. Long, and C. G. Will- cocks, Deep generative modelling: A comparative review of vaes, gans, normalizing flows, energy-based and au- toregressive models, IEEE transactions on pattern anal- ysis and machine intelligence44, 7327 (2021)

    2021

  3. [3]

    Sohl-Dickstein, E

    J. Sohl-Dickstein, E. Weiss, N. Maheswaranathan, and S. Ganguli, Deep Unsupervised Learning using Nonequi- librium Thermodynamics, inProceedings of the 3second International Conference on Mach. Learn., Proceedings of Mach. Learn. Research, Vol. 37, edited by F. Bach and D. Blei (Lille, France, 2015) pp. 2256–2265

    2015

  4. [4]

    J. Ho, A. Jain, and P. Abbeel, Denoising Diffusion Prob- abilistic Models, arXiv:2006.11239 (2020)

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  5. [5]

    Y. Song, J. Sohl-Dickstein, D. P. Kingma, A. Ku- mar, S. Ermon, and B. Poole, Score-Based Genera- tive Modeling through Stochastic Differential Equations, arXiv:2011.13456 (2021)

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  6. [6]

    B. D. Anderson, Reverse-time diffusion equation models, Stochastic Processes Appl.12, 313 (1982)

    1982

  7. [7]

    C. Chen, Q. Zhao, M. Zhou, Z. He, Z. Sun, and H. Situ, Quantum generative diffusion model: a fully quantum- mechanical model for generating quantum state ensem- ble, arXiv preprint arXiv:2401.07039 (2024)

    work page arXiv 2024

  8. [8]

    G. Kwun, B. Zhang, and Q. Zhuang, Mixed-state quan- tum denoising diffusion probabilistic model, Physical Re- view A111, 032610 (2025)

    2025

  9. [9]

    Parigi, S

    M. Parigi, S. Martina, and F. Caruso, Quantum-noise- driven generative diffusion models, Advanced Quantum Technologies8, 2300401 (2025)

    2025

  10. [10]

    R. Nasu, G. Tanaka, and A. Tsuchiya, Quantum re- versibility meets classical reverse diffusion, arXiv preprint arXiv:2510.18512 (2025)

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  11. [11]

    Zhang, P

    B. Zhang, P. Xu, X. Chen, and Q. Zhuang, Generative quantum machine learning via denoising diffusion prob- abilistic models, Physical Review Letters132, 100602 (2024)

    2024

  12. [12]

    X. Liu, J. Zhuang, W. Hou, and Y.-Z. You, Measurement-based quantum diffusion models, arXiv preprint arXiv:2508.08799 (2025)

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  13. [13]

    Stochastic Schr\"odinger Equations for Quantum Reverse Diffusion

    E. Gabbassov, Exact stochastic differential equa- tions for quantum reverse diffusion, arXiv preprint arXiv:2511.15919 (2025)

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  14. [14]

    See Supplemental Material at [URL] for more details

  15. [15]

    Belavkin, A stochastic posterior Schr¨ odinger equation for counting nondemolition measurement, Lett

    V. Belavkin, A stochastic posterior Schr¨ odinger equation for counting nondemolition measurement, Lett. Math. Phys.20, 85 (1990)

    1990

  16. [16]

    Gardiner and P

    C. Gardiner and P. Zoller,Quantum noise(Berlin, Hei- delberg, 2004)

    2004

  17. [17]

    H. P. Breuer and F. Petruccione,The theory of open quantum systems(Great Clarendon Street, 2002)

    2002

  18. [18]

    Lindblad, On the generators of quantum dynamical semigroups, Comm

    G. Lindblad, On the generators of quantum dynamical semigroups, Comm. Math. Phys.48, 119 (1976)

    1976

  19. [19]

    Gorini, A

    V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, Completely positive dynamical semigroups of N-level sys- tems, J. Math. Phys.17, 821 (1976)

    1976

  20. [20]

    Benoist, M

    T. Benoist, M. Fraas, Y. Pautrat, and C. Pellegrini, In- variant measure for stochastic schr¨ odinger equations, in Annales Henri Poincar´ e, Vol. 22 (Springer, 2021) pp. 347–374

    2021

  21. [21]

    Carollo, J

    F. Carollo, J. P. Garrahan, and R. L. Jack, Large Devia- tions at Level 2.5 for Markovian Open Quantum Systems: Quantum Jumps and Quantum State Diffusion, J. Stat. Phys.184, 13 (2021)

    2021

  22. [22]

    H. Kwon, R. Mukherjee, and M. S. Kim, Reversing Lind- blad Dynamics via Continuous Petz Recovery Map, Phys. Rev. Lett.128, 020403 (2022)

    2022

  23. [23]

    Petz, Sufficient subalgebras and the relative entropy of states of a von Neumann algebra, Comm

    D. Petz, Sufficient subalgebras and the relative entropy of states of a von Neumann algebra, Comm. Math. Phys. 105, 123 (1986)

    1986

  24. [24]

    A. J. Parzygnat and J. Fullwood, From time-reversal symmetry to quantum bayes’ rules, PRX Quantum4, 020334 (2023)

    2023

  25. [25]

    G. Bai, F. Buscemi, and V. Scarani, Quantum Bayes’ rule and Petz transpose map from the minimum change principle, arXiv:2410.00319 (2025)

    work page arXiv 2025

  26. [26]

    Maassen and B

    H. Maassen and B. K¨ ummerer, Purification of quan- tum trajectories, IMS Lecture Notes Monogr. Ser. , 252 (2006)

    2006

  27. [27]

    N. H. Amini, M. Bompais, and C. Pellegrini, On asymp- totic stability of quantum trajectories and their Cesaro 6 mean, J. Phys. A54, 385304 (2021)

    2021

  28. [28]

    Bompais, N

    M. Bompais, N. H. Amini, J. P. Garrahan, and M. Gut ¸˘ a, The rate of purification of quantum trajecto- ries, arXiv:2601.14023 (2026)

    work page arXiv 2026

  29. [29]

    Guilmin, P

    P. Guilmin, P. Rouchon, and A. Tilloy, Correlation functions for realistic continuous quantum measurement, IFAC-PapersOnLine56, 5164 (2023)

    2023

  30. [30]

    Grigoletto, C

    T. Grigoletto, C. Pellegrini, and F. Ticozzi, Quantum model reduction for continuous-time quantum filters: T. grigoletto et al., inAnnales Henri Poincar´ e(Springer,

  31. [31]

    Biroli, T

    G. Biroli, T. Bonnaire, V. de Bortoli, and M. M´ ezard, Dy- namical Regimes of Diffusion Models, arXiv:2402.18491 (2024)

    work page arXiv 2024

  32. [32]

    Kadkhodaie, F

    Z. Kadkhodaie, F. Guth, E. P. Simoncelli, and S. Mallat, Generalization in diffusion models arises from geometry- adaptive harmonic representations, arXiv:2310.02557 (2024)

    work page arXiv 2024

  33. [33]

    Memorisation, convergence and generalisation in generative models

    A. Maillard and S. Goldt, Memorisation, convergence and generalisation in generative models, arXiv:2605.21402 (2026)

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  34. [34]

    Ikeda and S

    N. Ikeda and S. Watanabe,Stochastic differential equa- tions and diffusion processes, Vol. 24 (Elsevier, 2014). 7 Supplementary Material for Generating quantum ensembles via reverse-time quantum diffusions I. GENERATOR OF THE FORWARD EVOLUTION The generator and adjoint generator of the quantum diffusion are the following (cf. [20]). For any scalar smooth ...

    2014

  35. [35]

    Then the following are equivalent:

    Characterisation of commutator directions Letψ=|ψ⟩⟨ψ|be a rank-one projector, so thatψ 2 =ψ, and letX(ψ)∈ B(H) be Hermitian. Then the following are equivalent:

  36. [36]

    Proof.(1)⇒(2)

    There exists a Hermitian operatorH(ψ) such that X(ψ) =−i[H(ψ), ψ].(78) 2.X(ψ) satisfies ψX(ψ)ψ= 0,(1−ψ)X(ψ)(1−ψ) = 0.(79) a. Proof.(1)⇒(2). AssumeX(ψ) =−i[H(ψ), ψ]. Then ψX(ψ)ψ=−i ψ H(ψ)ψ−ψH(ψ) ψ(80) =−i ψH(ψ)ψ−ψH(ψ)ψ = 0,(81) where we usedψ 2 =ψ. Moreover, (1−ψ)X(ψ)(1−ψ) =−i(1−ψ) H(ψ)ψ−ψH(ψ) (1−ψ) (82) = 0,(83) since (1−ψ)ψ=ψ(1−ψ) = 0. (2)⇒(1). Assume ψX...

  37. [37]

    Verification of the conditionψX(ψ)ψ= 0 Recall that the backward drift has been written in the form D(ψ) +X(ψ),(97) with X(ψ) =i[H, ψ]−2D(ψ) + divD(ψ) +D(ψ)∇logµ T−t (ψ).(98) We now check that ψX(ψ)ψ= 0 (99) for every pure stateψ. Sinceψis a pure state, it is a rank-one orthogonal projection, so that ψ2 =ψ,tr(ψ) = 1, ψAψ= tr(Aψ)ψfor every operatorA.(100) W...

  38. [38]

    Verification of the condition(1−ψ)X(ψ)(1−ψ) = 0 We now verify the second condition (1−ψ)X(ψ)(1−ψ) = 0.(161) Recall that X(ψ) =i[H, ψ]−2D(ψ) + divD(ψ) +D(ψ)∇logµ T−t (ψ),(162) where D(ψ) = MX m=1 LmψL† m − 1 2 {L† mLm, ψ} ,(163) and D(ψ)∇logµ T−t (ψ) = MX m=1 Km(ψ)S m(ψ, t),(164) with Km(ψ) =L mψ+ψL † m −tr((L m +L † m)ψ)ψ.(165) For simplicity, let us set ...