A 2025 result (Nasu, Tanaka, Tsuchiya, arXiv:2510.18512) showed that a semiclassical Wigner–Moyal expansion of the Lindblad equation, truncated at first order in ℏ\hbar, reduces it to a Fokker–Planck equation for the Wigner function, and that applying the same expansion to the Petz-reversed Lindblad equation yields exactly the Bayes reverse-time diffusion equation underlying score-based generative models — a direct phase-space identification of the quantum Petz map with Bayes' rule. The demonstrated correspondence is, however, leading-order only, silently treats the Wigner functions of the state and of the reference state as probability densities, and is confined to Markovian semigroup dynamics. Completing the correspondence requires: (a) determining the O(ℏ2)O(\hbar^2) and higher-order corrections to both reduced equations, whether the Bayes form of the reverse drift survives at higher order, and under what conditions the expansion is controlled; (b) settling whether the probabilistic reverse-diffusion interpretation remains consistent when the state's or reference's Wigner function takes negative values — the reverse drift involves ∇log⁡W(γ)\nabla\log W^{(\gamma)} and becomes ill-defined for negative W(γ)W^{(\gamma)} — and how the correspondence must then be modified; (c) extending it beyond the assumed semiclassical Markovian regime (mesoscopic systems, non-Markovian dynamics not described by a Lindblad semigroup); and (d) constructing concrete quantum analogues of diffusion-based generative models that exploit the correspondence. Recent 2026 results show that naively lifting the classical-looking reverse equation to an exact Gaussian quantum semigroup violates complete positivity once nonclassicality exceeds a sharp threshold, so the Gaussian corner of question (b) is sharply narrowed but the general problem — and the higher-order, mesoscopic, non-Markovian, and constructive parts — remains open.

Background

Open quantum systems whose evolution enjoys the Markov property are described by the Lindblad equation, the generator of a one-parameter semigroup of trace-preserving completely positive (TPCP) maps, i.e. maps taking density operators to density operators even when the system is entangled with an unobserved ancilla. The phase-space description of quantum mechanics uses the Wigner function, the Weyl transform of the density operator: a real, normalized distribution on 2N2N-dimensional phase space that is not guaranteed to be nonnegative and is therefore called a quasiprobability. Operator products transform to the noncommutative star (Moyal) product, which is a formal power series in ℏ\hbar; truncating that series is the standard semiclassical (truncated-Wigner-type) approximation. For a Lindblad equation whose Hamiltonian and Lindblad operators have ℏ\hbar-independent Weyl symbols (their representative phase-space functions under the Wigner transform), expanding the Wigner transform of the equation up to O(ℏ)O(\hbar) yields a Fokker–Planck equation — the diffusion equation for the probability density of a classical stochastic process with drift vector fμf^\mu and diffusion matrix GμνG^{\mu\nu} — for the Wigner function, with GμνG^{\mu\nu} automatically positive semidefinite, so that a nonnegative Wigner function stays nonnegative.

Classical reverse diffusion: applying Bayes' rule to the Fokker–Planck equation produces a reverse-time diffusion equation whose drift is minus the forward drift plus the diffusion matrix contracted with the gradient of the log-density of the terminal (reference) distribution; this Bayes reverse process is exactly what modern score-based generative models sample, with the log-density gradient known as the score function.

Quantum reversibility: the canonical reversal of a quantum channel with respect to a reference state γ^\hat\gamma is the Petz (transpose) map, long discussed as the quantum analogue of Bayes' rule, and continuous-time Lindblad dynamics can be reversed by a Petz map that is itself generated by a reversed Lindblad equation whose Hamiltonian and Lindblad operators are built from the running reference state γ^t1/2\hat\gamma_t^{1/2} (Kwon, Mukherjee and Kim, Phys. Rev. Lett. 128, 020403 (2022)).

The source paper (Nasu, Tanaka, Tsuchiya, arXiv:2510.18512) proved that applying the same O(ℏ)O(\hbar) Wigner–Moyal approximation to that reversed Lindblad equation produces precisely the Bayes reverse-time diffusion equation, with reverse drift fˉμ=−fμ+1W(γ)∂ν(GμνW(γ))\bar f^\mu = -f^\mu + \frac{1}{W^{(\gamma)}}\partial_\nu(G^{\mu\nu}W^{(\gamma)}), where W(γ)W^{(\gamma)} is the Wigner function of the reference state; this is the divergence form of the Bayes reverse drift, equal to the score form −fμ+Gμν∂νlog⁡W(γ)-f^\mu + G^{\mu\nu}\partial_\nu\log W^{(\gamma)} plus the extra term ∂νGμν\partial_\nu G^{\mu\nu}, the two coinciding when the diffusion matrix GμνG^{\mu\nu} is independent of the phase-space point: a direct phase-space identification of the Petz map with Bayes' rule, unifying quantum reversibility with classical reverse diffusion. The authors also construct WKB-type (Wentzel-Kramers-Brillouin, i.e. exponential-ansatz) solutions of the two reduced equations.

The demonstrated correspondence has three structural gaps. First, the derivation truncates the star-product (Moyal) expansion at O(ℏ)O(\hbar): the size, structure, and physical consequences of the neglected higher-order terms — and whether the Bayes form of the reverse drift survives once they are included — were left undetermined, as were the conditions (on states, Lindblad-operator classes, and time scales) under which the expansion is controlled. Second, the probabilistic reading of the reduced equations assumes that the Wigner functions of the state and of the reference act as probability densities; the source notes that positivity of GμνG^{\mu\nu} preserves the probabilistic interpretation for initially nonnegative Wigner functions, but Wigner functions can be negative, and the reverse drift contains ∇log⁡W(γ)\nabla\log W^{(\gamma)}, which is ill-defined as a real vector field wherever W(γ)<0W^{(\gamma)}<0. Third, the construction requires Markovian Lindblad-semigroup dynamics and an ℏ\hbar expansion, so genuinely mesoscopic regimes and non-Markovian dynamics lie outside its scope; the authors name the mesoscopic extension, and its use for constructing quantum analogues of diffusion-based generative models, as future directions.

Recent adjacent results sharpen the stakes. Fayad (arXiv:2603.06488, arXiv:2603.19024, 2026) showed that when the forward dynamics is an exact Gaussian quantum semigroup (quadratic Hamiltonian, linear Lindblad operators), whose Wigner function obeys an exact Fokker–Planck equation, extracting the Bayes reverse drift at fixed diffusion and lifting it to a Gaussian quantum reverse semigroup violates complete positivity exactly when cosh⁡(2r)>ν\cosh(2r)>\nu, where rr is the squeezing parameter (setting the phase-dependent noise anisotropy of the Gaussian reference state) and ν\nu its thermal occupation number: any Gaussian completely-positive repair must inject extra reverse diffusion (with a closed-form minimum cost), and exact reversal of a pure nonclassical endpoint is dynamically unattainable. This quantifies, in the Gaussian class, precisely what the neglected higher-order quantum terms must repair, and shows the naive classical-looking reverse equation cannot be taken at face value once nonclassicality is significant. In parallel, a trajectory-level quantum reverse-diffusion line has developed (Gabbassov, arXiv:2511.15919; Bompais, Guță and Garrahan, arXiv:2606.03848; Dubey and John, arXiv:2604.21210) using monitored quantum trajectories with a feedback Hamiltonian playing the role of the score; it does not use or extend the phase-space Petz correspondence at issue here.

Problem Statement

Establish the complete structure and domain of validity of the phase-space correspondence between the Petz map (quantum reversibility) and Bayes' rule (classical reverse diffusion) demonstrated to leading order in ℏ\hbar by Nasu, Tanaka and Tsuchiya. The common objective is to turn the leading-order formal identity into a complete, physically consistent, and operationally usable theory of quantum reverse diffusion; the four components below are formulated in this statement as the completion program of the demonstrated result and are scientifically linked by that single objective.

(i) Higher-order structure and controlledness. Derive the O(ℏ2)O(\hbar^2) (and, if possible, all-order) corrections to both reduced equations — the Fokker–Planck equation obtained from the Lindblad equation and the reverse-time diffusion equation obtained from the Petz-reversed Lindblad equation — determine whether the Bayes form of the reverse drift (in the source's reverse-time diffusion equation, fˉμ=−fμ+1W(γ)∂ν(GμνW(γ))\bar f^\mu = -f^\mu + \frac{1}{W^{(\gamma)}}\partial_\nu(G^{\mu\nu}W^{(\gamma)}), equivalent to the score form −fμ+Gμν∂νlog⁡W(γ)-f^\mu + G^{\mu\nu}\partial_\nu\log W^{(\gamma)} up to the ∂νGμν\partial_\nu G^{\mu\nu} term) survives once the neglected star-product (Moyal) terms are included, and state the conditions (on the state, the reference state, the Weyl-symbol class of the Hamiltonian and Lindblad operators, and time scales) under which the expansion is controlled, i.e. the neglected terms remain small.

(ii) Negativity domain. Settle whether the probabilistic reverse-diffusion interpretation of the correspondence remains consistent when the Wigner functions W(ρ)W^{(\rho)} of the state or W(γ)W^{(\gamma)} of the reference take negative values — noting that the reverse drift involves 1/W(γ)1/W^{(\gamma)} and hence the log-gradient ∇log⁡W(γ)\nabla\log W^{(\gamma)}, which is ill-defined as a real vector field wherever W(γ)≤0W^{(\gamma)}\le 0 — by proving a domain-of-validity theorem for a class of state/reference/Lindblad triples, or by exhibiting and quantifying the obstruction (a counterexample within the source's assumptions, or a repair with its cost). The known Gaussian fixed-diffusion no-go — complete positivity is violated when cosh⁡(2r)>ν\cosh(2r)>\nu upon lifting the classical-looking reverse drift to an exact Gaussian quantum reverse semigroup — supplies a benchmark in that subclass: an answer to (ii) whose framework covers such exact Gaussian lifts must reproduce that obstruction.

(iii) Regime extension. Extend the correspondence beyond the assumed regime: to mesoscopic systems where the semiclassical expansion breaks down, and to non-Markovian dynamics not described by a Lindblad semigroup (where the channel-level Petz map does not directly apply, so that an appropriate notion of reversal must first be formulated), in each case formulating precisely in what sense the Petz–Bayes identification survives and proving it.

(iv) Constructive exploitation. Construct concrete quantum analogues of diffusion-based generative models that exploit this Petz–Bayes correspondence: an architecture and a training/sampling procedure operationalizing the reversed Wigner dynamics, including the operational meaning of sampling from Wigner-like quasiprobabilities that may be negative, with the implemented dynamics completely positive.

A complete answer discharging (i)–(iv) closes the completion program stated here; partial answers (e.g. the Gaussian-class no-go already in the literature) are measured against their component. The author-declared decoherence/classicalization connection is an adjacent direction outside this program.

References

  1. R. Nasu, G. Tanaka, A. Tsuchiya, "Quantum Reversibility Meets Classical Reverse Diffusion", arXiv:2510.18512 (2025; v2 2026), https://arxiv.org/abs/2510.18512
  2. H. Kwon, R. Mukherjee, M. S. Kim, "Reversing Lindblad Dynamics via Continuous Petz Recovery Map", Phys. Rev. Lett. 128, 020403 (2022), DOI 10.1103/PhysRevLett.128.020403, https://doi.org/10.1103/PhysRevLett.128.020403
  3. G. Bai, F. Buscemi, V. Scarani, "Quantum Bayes' Rule and Petz Transpose Map from the Minimum Change Principle", Phys. Rev. Lett. 135, 090203 (2025), DOI 10.1103/5n4p-bxhm, https://doi.org/10.1103/5n4p-bxhm
  4. B. D. O. Anderson, "Reverse-time diffusion equation models", Stochastic Processes and their Applications 12, 313-326 (1982), DOI 10.1016/0304-4149(82)90051-5, https://doi.org/10.1016/0304-4149(82)90051-5
  5. Y. Song, J. Sohl-Dickstein, D. P. Kingma, A. Kumar, S. Ermon, B. Poole, "Score-Based Generative Modeling through Stochastic Differential Equations", arXiv:2011.13456 (2020), https://arxiv.org/abs/2011.13456
  6. A. Fayad, "Score Reversal Is Not Free for Quantum Diffusion Models", arXiv:2603.06488 (2026), https://arxiv.org/abs/2603.06488
  7. A. Fayad, "Exact Law of Quantum Reversibility under Gaussian Pure Loss", arXiv:2603.19024 (2026), https://arxiv.org/abs/2603.19024
  8. E. Gabbassov, "Stochastic Schrödinger equations for quantum reverse diffusion", Phys. Rev. Research 8, 023329 (2026), DOI 10.1103/f7fd-2rb9, arXiv:2511.15919, https://doi.org/10.1103/f7fd-2rb9
  9. M. Bompais, M. Guță, J. P. Garrahan, "Generating quantum ensembles via reverse-time quantum diffusions", arXiv:2606.03848 (2026), https://arxiv.org/abs/2606.03848
  10. S. Dubey, A. John, "The Feedback Hamiltonian is the Score Function: A Diffusion-Model Framework for Quantum Trajectory Reversal", arXiv:2604.21210 (2026), https://arxiv.org/abs/2604.21210
  11. A. Cacioppo, L. Colantonio, S. Bordoni, S. Giagu, "Quantum Diffusion Models", arXiv:2311.15444 (2023), https://arxiv.org/abs/2311.15444