Quantum Reversibility Meets Classical Reverse Diffusion

2025/10/21 by Ryota Nasu, Nasu, Ryota, Gota Tanaka +3 · 1 citation
Physics and Astronomy · #FOS: Physical sciences #Quantum Mechanics and Applications #Quantum Physics (quant-ph) #Statistical Mechanics and Entropy #stochastic dynamics and bifurcation

paper · pdf · doi:10.48550/arxiv.2510.18512

openalex publication_date 2025/10/21 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28

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.

Citations

• Quantum Flow Matching
• Measurement-Based Quantum Diffusion Models
• Path-Integral Formulation of Bosonic Markovian Open Quantum Dynamics with Monte Carlo stochastic trajectories using the Glauber-Sudarshan P, Wigner, and Husimi Q Functions and Hybrids
• Beyond Equilibrium: Non-Equilibrium Foundations Should Underpin Generative Processes in Complex Dynamical Systems
• Mixed-State Quantum Denoising Diffusion Probabilistic Model
• Quantum Bayes' rule and Petz transpose map from the minimum change principle
• Path-Integral Formulation of Truncated Wigner Approximation for Bosonic Markovian Open Quantum Systems
• Quantum State Generation with Structure-Preserving Diffusion Model
• Quantum Generative Diffusion Model: A Fully Quantum-Mechanical Model for Generating Quantum State Ensemble
• Quantum Denoising Diffusion Models
• Quantum Diffusion Models
• Score-Based Generative Modeling through Stochastic Differential Equations
• Generative Modeling by Estimating Gradients of the Data Distribution
• Deep Unsupervised Learning using Nonequilibrium Thermodynamics
• Reversing quantum dynamics with near-optimal quantum and classical fidelity
• Completely positive dynamical semigroups of N-level systems

Cited by

• Exact Stochastic Differential Equations for Quantum Reverse Diffusion

Related

• Quantum Mechanics as a Reversible Diffusion Theory
• Classical correspondence beyond the Ehrenfest time for open quantum systems with general Lindbladians
• Exact Stochastic Differential Equations for Quantum Reverse Diffusion
• Measurement-Based Quantum Diffusion Models
• Stochastic Theory of Quantum Mechanics