Quantum Reversibility Meets Classical Reverse Diffusion

arXiv:2510.18512 · quant-ph · Submitted 2025-10-21 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: We're looking at the paper: "Quantum Reversibility Meets Classical Reverse Diffusion".

Mira: Here's the gist of the paper. The gist The authors demonstrate that applying a semiclassical approximation to both the Lindblad equation and its reversal yields equations for the Wigner function that coincide, establishing a direct correspondence between quantum reversibility via the Petz map and classical reverse diffusion via Bayes’ rule.

Kai: Correspondence between Quantum Reversibility and Classical Reverse Diffusion The core finding of this work is that applying a semiclassical approximation to the Lindblad equation associated with the Petz map produces an equation for the Wigner function that coincides with that obtained from the Fokker-Planck equation via Bayes’ rule, which underlies modern diffusion-based generative modelsThis finding reveals a direct correspondence between the Petz map and Bayes’ rule, highlighting a unified view of quantum reversibility and classical reverse diffusion Derivation via Semiclassical Approximation The paper establishes this link by considering a semiclassical approximation of the Lindblad equation that reduces to the Fokker-Planck equation for the Wigner function, which is a quasiprobability distribution defined on phase space as the Wigner transform of the density operatorThe derivation involves expanding both equations up to order O(ħ) using Eqs.

Mira: (3.1) and (3.2)This process shows that the reversed Lindblad equation reduces to the reverse-time diffusion equation under the semiclassical approximation Construction of WKB Solutions The authors construct WKB solutions for both the semi-classical Lindblad and its reversal semi-classical equation, examining a concrete example in appendix BFor the forward process, the solution for the zeroth-order term is given by W(ρ)t,0 = w0(x, t)e−2 f tThe second-order term in the expansion of W(ρ)t,0 shows that ∂2∂Q2W(ρ)t,0 is related to the initial conditions of the characteristic curveFor the reverse process, Eq.

Kai: (3.9) takes the form of the reverse-time diffusion equation (2.10)The zeroth-order solution for the reverse process is given by W¯(ρ)0(Q¯, P¯, t¯) = w¯ 0(Q¯, P̄, t̄)e−f t̄ Reconstruction of Dynamics The analysis of the reversed Lindblad operator and Hamiltonian yields expressions for the reversed drift terms that encode information about the forward processThe drift in the reverse process is given by ¯fµ(x, t) = − fµ(x, t) + 1/W(γ)t Xν ∂/∂xν Gµν(x, t)W(γ)tThis structure is what drives the distribution toward the initial state P(x, 0) Implications and Outlook The paper provides a new insight into the connection between quantum reversibility and classical reverse diffusion, linking the Petz map with Bayes’ rulePossible future directions include investigating a relationship of our findings with decoherence, which can be viewed as certain classicalizationAnother direction is extending the correspondence found in this paper to mesoscopic systems and exploring its implications for constructing quantum analogues of diffusion-based generative modelsThe work also shows that for N = 1, Hˆ = 0 and Lˆ = pˆ, the Lindblad and reversal Lindblad equations are closed for coherent states and exactly correspond to the diffusion and reversal diffusion equations, respectively.

Paper summary: Kai: So we're looking at this paper today, "Quantum Reversibility Meets Classical Reverse Diffusion." Basically, it connects how time runs backward in the quantum world with how things run backward in classical physics.

Mira: Exactly. The main idea here is that there's a direct link between the Petz map, which is this quantum reversal process we're looking at, and Bayes’ rule, which is what underpins those diffusion models we see today for generating data.

Lev: So it’s about showing that if you take the Lindblad equation describing quantum evolution and apply a semiclassical approximation to it, you get an equation for the Wigner function that lines up perfectly with what you get from applying Bayes’ rule to the Fokker-Planck equation, which is the classical counterpart.

Kai: That sounds a bit abstract. What does this mean for someone who isn't deep into quantum dynamics or classical diffusion models?

Mira: It means we have a way to bridge those two worlds using this semiclassical approximation, which simplifies things down to the Fokker-Planck equation for the Wigner function, which is like a probability distribution in phase space.

Lev: And what it changes for real hardware is that we can see how the reversed Lindblad equation reduces exactly to that reverse-time diffusion equation when you use this semiclassical approximation, which helps us understand the dynamics better.

Kai: So we're looking at a correspondence between quantum reversibility via the Petz map and classical reverse diffusion via Bayes’ rule. Is that what they're claiming?

Mira: That’s the core finding of this work, establishing a direct correspondence between the Petz map and Bayes’ rule, which unifies quantum reversibility with classical reverse diffusion.

Lev: From an engineering standpoint, it means we can use the structure of the reversed Lindblad operator and Hamiltonian to get expressions for the reversed drift terms that actually encode information about how things went forward in time.

Kai: So they’re saying this structure drives the distribution toward its initial state P(x, zero) when you reverse the process?

Mira: Right. The drift in the reverse process is given by mu(x, t) as - f mu(x, t) + one/W(gamma) t X nu d/d x nu G mu nu(x, t)W(gamma) t <ref:2510.18512#pg1>.

Lev: That formula shows exactly how the structure of the forward process dictates the reverse drift, which is what drives it toward that initial distribution.

Kai: So before we get into how they prove this, just to recap, this paper is about showing that a semiclassical approximation applied to both equations—the Lindblad and its reversal—yields matching equations for the Wigner function.

Mira: And that these two resulting equations are linked through the application of Bayes’ rule to the classical Fokker-Planck equation.

Lev: That's a pretty neat structural alignment between quantum dynamics and classical stochastic processes when you look at them through this approximation.

Kai: So, what’s the next part of this discussion? Where do we go from here with this connection?

Mira: The authors spend a lot of time constructing WKB solutions for both the semi-classical Lindblad equation and its reversal semi-classical equation, looking at a concrete example in appendix B.

Lev: They look at the zeroth-order term for the forward process, which is given by W(rho) t,zero = w zero(x, t)e-two f t.

Kai: And they also look at the reverse process with its zeroth-order solution for W zero(Q, P,) being w(Q, P,)e-f.

Mira: They then show how the second-order term in the expansion of the forward solution relates to the initial conditions of a characteristic curve.

Lev: For the reverse process, equation three point nine takes on this form of W zero(Q, P,) = w(Q, P,)e-f.

Kai: It’s interesting how they build these solutions up to order O to see this correspondence.

Mira: They show that applying the same approximation to the Lindblad equation associated with the Petz map produces an equation that coincides with what you get from the Fokker-Planck equation via Bayes’ rule.

Lev: That’s a pretty strong result because it directly connects a specific quantum map, the Petz map, to a classical statistical tool, Bayes' rule.

Kai: So we have this unified view of quantum reversibility and classical reverse diffusion established through this semiclassical approximation.

Mira: It provides a new way to think about how information flows when you reverse a dynamic process in both the quantum and classical regimes.

Lev: The paper also points out that for the specific case where N=one the Hamiltonian is zero, and the Lindblad operator is just, both equations close exactly for coherent states <ref:2510.18512#pg1>.

Kai: That’s a good detail for anyone working with simpler quantum systems; it shows an exact correspondence there.

Mira: The authors also mention some future directions, like investigating how this correspondence relates to decoherence, which can be seen as a way to classicalize things.

Lev: They also suggest extending this correspondence to mesoscopic systems and exploring what that means for building quantum analogues of those diffusion-based generative models.

Kai: So the paper does a lot of groundwork on showing this link through the semiclassical approximation, and it opens up some avenues for how we might use these ideas in other areas.

Mira: It really shows how deeply classical concepts like reverse diffusion are baked into the structure of quantum reversibility when you look at things through this specific mathematical lens.

Conclusion: Kai: So, we’ve been looking at how the quantum evolution described by the Lindblad equation connects to classical reverse diffusion through this Petz map thing.

Mira: That’s right, and this paper shows that when you use a semiclassical approximation on both sides, you get matching equations for the Wigner function.

Lev: From an engineering standpoint, it means we can see how the reversed Lindblad operator reduces exactly to the reverse-time diffusion equation under those same approximations.

Kai: What does this mean for the big picture? It’s about showing a direct link between quantum reversibility and classical reverse diffusion.

Mira: Exactly, it unifies these two concepts in a way that was previously quite separate.

Lev: For real hardware, it suggests we have a mathematical structure to guide the dynamics toward an initial state when you reverse the process.

Kai: It's about building that understanding of how information flows backward in both quantum and classical systems, right?

Mira: That’s it. The authors are really connecting the Petz map to Bayes’ rule in a way that bridges those two domains.

Lev: But they also point out where the method stops working, so we know what's not covered by this specific approximation.

Kai: So they laid out this correspondence, and now we need to see how it applies practically.

Department of Electrical and Computer Engineering, National Institute of Technology, Wakayama College · Institute for Mathematical Informatics, Meiji Gakuin University · Department of Physics, Shizuoka University

quant-ph

Submitted: 2025-10-21

Updated: 2026-10-22

Comments: 12+16 pages, 1 figure, This is published version

Journal ref: J. Phys. Soc. Jpn., Vol.95, No.10, Article ID: 104004, 2026

DOI: 10.7566/JPSJ.95.104004

License: http://creativecommons.org/licenses/by-nc-nd/4.0/

Importance score: 76/100

The gist: The gist The authors demonstrate that applying a semiclassical approximation to both the Lindblad equation and its reversal yields equations for the Wigner function that coincide, establishing a

Key concepts

Lindblad Equation: This equation describes how a quantum system evolves over time when it interacts with its environment. It governs Markovian quantum dynamics and is the fundamental equation used to model open quantum systems.
Petz Map: The Petz map is a mathematical tool related to the equality condition for the data-processing inequality in quantum relative entropy. It serves as a key connection point between forward and reverse processes in quantum dynamics.
Wigner Function: The Wigner function is a quasiprobability distribution defined on phase space. It acts as the classical analogue of the density operator, allowing quantum evolution to be described using tools from classical statistical mechanics.

Terminology

Generative Model — 1× in this paper · explained in 2 episodes
Generative Models — 1× in this paper · explained in 3 episodes
Wigner function — 1× in this paper · explained in 3 episodes

Summary

The gist The authors demonstrate that applying a semiclassical approximation to both the Lindblad equation and its reversal yields equations for the Wigner function that coincide, establishing a direct correspondence between quantum reversibility via the Petz map and classical reverse diffusion via Bayes’ rule.

Correspondence between Quantum Reversibility and Classical Reverse Diffusion

The core finding of this work is that applying a semiclassical approximation to the Lindblad equation associated with the Petz map produces an equation for the Wigner function that coincides with that obtained from the Fokker-Planck equation via Bayes’ rule, which underlies modern diffusion-based generative modelsThis finding reveals a direct correspondence between the Petz map and Bayes’ rule, highlighting a unified view of quantum reversibility and classical reverse diffusion

Derivation via Semiclassical Approximation

The paper establishes this link by considering a semiclassical approximation of the Lindblad equation that reduces to the Fokker-Planck equation for the Wigner function, which is a quasiprobability distribution defined on phase space as the Wigner transform of the density operatorThe derivation involves expanding both equations up to order O(ħ) using Eqs. (3.1) and (3.2)This process shows that the reversed Lindblad equation reduces to the reverse-time diffusion equation under the semiclassical approximation

Construction of WKB Solutions

The authors construct WKB solutions for both the semi-classical Lindblad and its reversal semi-classical equation, examining a concrete example in appendix BFor the forward process, the solution for the zeroth-order term is given by W(ρ)t,0 = w0(x, t)e−2 f tThe second-order term in the expansion of W(ρ)t,0 shows that ∂2∂Q2W(ρ)t,0 is related to the initial conditions of the characteristic curveFor the reverse process, Eq. (3.9) takes the form of the reverse-time diffusion equation (2.10)The zeroth-order solution for the reverse process is given by W¯(ρ)0(Q¯, P¯, t¯) = w¯ 0(Q¯, P̄, t̄)e−f t̄

Reconstruction of Dynamics

The analysis of the reversed Lindblad operator and Hamiltonian yields expressions for the reversed drift terms that encode information about the forward processThe drift in the reverse process is given by ¯fµ(x, t) = − fµ(x, t) + 1/W(γ)t Xν ∂/∂xν Gµν(x, t)W(γ)tThis structure is what drives the distribution toward the initial state P(x, 0)

Implications and Outlook

The paper provides a new insight into the connection between quantum reversibility and classical reverse diffusion, linking the Petz map with Bayes’ rulePossible future directions include investigating a relationship of our findings with decoherence, which can be viewed as certain classicalizationAnother direction is extending the correspondence found in this paper to mesoscopic systems and exploring its implications for constructing quantum analogues of diffusion-based generative modelsThe work also shows that for N = 1, Hˆ = 0 and Lˆ = pˆ, the Lindblad and reversal Lindblad equations are closed for coherent states and exactly correspond to the diffusion and reversal diffusion equations, respectively. The authors have constructed WKB solutions for the semi-classical Lindblad and reverse semi-classical Lindblad equations. The paper was supported in part by JSPS KAKENHI Grant Numbers 21K03532 and 25K07319. The work is available as arXiv:quant-ph/quant-ph>The paper is organized as follows. In section 2, we review the Petz map and the Lindblad equationIn section 3, we first see how a semiclassical approximation reduces the Lindblad equation to the Fokker-Planck equation for the Wigner function, and then show that the same approximation of the Lindblad equation with the Petz map yields a diffusion equation that is obtained by applying Bayes’ rule to the Fokker-Planck equationIn section 4, we construct WKB solutions of the semi-classical Lindblad equation and its reversal semi-classical equationSection 5 is devoted to conclusion and outlook. The Petz map was originally introduced in Ref. 1) as the equality condition for the data-processing inequality of the quantum relative entropy, and later rediscovered in the contexts of quantum error correction and statistical physicsThe Lindblad equation describes the most general form of Markovian quantum dynamics as a one-parameter semigroup of TPCP mapsIt has been shown in Refs. 6, 7) that this Petz map leads to a reversed Lindblad equationThe authors would like to thank Tatsuro Yuge and Kazuki Kobayashi for discussions. A.T. was supported in part by JSPS KAKENHI Grant Numbers 21K03532 and 25K07319. The paper is organized as follows. In section 4, we construct WKB solutions for the semi-classical Lindblad equation (3.4) (section 4.1) and its reversal semi-classical equation (3.9) (section 4.2). The authors are available at nasu@wakayama-nct.ac.jp, gotanak@mi.meijigakuin.ac.jp, and tsuchiya.asato@shizuoka.ac.jp. The paper is organized as follows: In section 2, we review the Petz map and the Lindblad equation (section 2.1) and on the Fokker-Planck equation and reverse diffusion process (section 2.2). In section 3, we first see how a semiclassical approximation reduces the Lindblad equation to the Fokker-Planck equation for the Wigner function, and then show that the same approximation of the Lindblad equation with the Petz map yields a diffusion equation that is obtained by applying Bayes’ rule to the Fokker-Planck equation. In section 4, we construct WKB solutions of the semi-classical Lindblad equation and its reversal semi-classical equation. Section 5 is devoted to conclusion and outlook. The authors are available at nasu@wakayama-nct.ac.jp, gotanak@mi.meijigakuin.ac.jp, and tsuchiya.asato@shizuoka.ac.jp. The paper is organized as follows: In section 2, we review the Petz map and the Lindblad equation (section 2.1) and on the Fokker-Planck equation and reverse diffusion process (section 2.2). In section 3, we first see how a semiclassical approximation reduces the Lindblad equation to the Fokker-Planck equation for the Wigner function, and then show that the same approximation of the Lindblad equation with the Petz map yields a diffusion equation that is obtained by applying Bayes’ rule to the Fokker-Planck equation. In section 4, we construct WKB solutions of the semi-classical Lindblad equation and its reversal semi-classical equation. Section 5 is devoted to conclusion and outlook. The Petz map was originally introduced in Ref.

Improvements for AI systems

Quantum Reversibility Model Integration: The AI system can now incorporate a direct correspondence between quantum dynamics and classical reverse diffusion by applying Bayes’ rule to the Lindblad equation associated with the Petz map. This allows for generating generative models that explicitly model quantum reversibility, unifying classical reverse diffusion with quantum dynamics.
Semiclassical Approximation for Wigner Function Evolution: The AI can utilize the semiclassical approximation of the Lindblad equation to evolve Wigner functions under both forward and reverse dynamics. This capability enables the system to simulate phase space evolution using equations like Eq (3.4) and its reversal, allowing for a more detailed understanding of quantum-to-classical transitions.
Score-Based Diffusion Model Enhancement: The system can be enhanced by incorporating drift terms derived from the Wigner function gradient, as shown in Eq (4.28). This allows the generative model to drive distributions toward the initial state P(x, 0), providing a theoretical foundation for score-based diffusion models with an explicit connection to quantum reversibility.
WKB Solution Construction for Generative Models: The AI can construct WKB solutions for both the semi-classical Lindblad and reverse semi-classical equations. This allows the system to explore specific regimes, such as those described in Appendix B, to generate precise solutions for complex quantum models, potentially leading to more accurate generative models.

Abstract

Bayes' rule connects forward and reverse processes in classical probability theory, and its quantum analogue has been discussed in terms of the Petz (transpose) map. For quantum dynamics governed by the Lindblad equation, the corresponding Petz map can also be written in Lindblad form. In classical stochastic systems, the analogue of the Lindblad equation is the Fokker-Planck equation, and applying Bayes' rule to it yields the reverse diffusion equation underlying modern diffusion-based generative models. It is known that a semiclassical approximation of the Lindblad equation yields the Fokker-Planck equation for the Wigner function, which is a quasiprobability distribution defined on phase space as the Wigner transform of the density operator. Here we demonstrate that applying the same approximation to the Lindblad equation associated with the Petz map produces an equation that coincides with that obtained from the Fokker-Planck equation via Bayes' rule. This finding establishes a direct correspondence between the Petz map and Bayes' rule, unifying quantum reversibility with classical reverse diffusion.