Quantum Reversibility Meets Classical Reverse Diffusion

TL;DR

This paper establishes a connection between quantum reversibility, via the Petz map, and classical reverse diffusion processes, using semiclassical approximations of the Lindblad equation and the Fokker-Planck equation for the Wigner function.

Contribution

It demonstrates that a semiclassical approximation of the Lindblad equation for the Petz map reproduces classical reverse diffusion equations, unifying quantum and classical probabilistic reversibility.

Findings

Petz map can be expressed in Lindblad form for quantum dynamics.

Semiclassical approximation of Lindblad yields Fokker-Planck for Wigner function.

Petz map corresponds to classical Bayes' rule in phase space.

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. Here we demonstrate that a semiclassical approximation of the Lindblad equation yields the Fokker-Planck equation for the Wigner function -- a quasiprobability distribution defined on phase space as the Wigner transform of the density operator. 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.