Score Reversal Is Not Free for Quantum Diffusion Models

TL;DR

This paper investigates the fundamental limits of quantum reverse diffusion, revealing a phase transition in noiselessness and establishing a benchmark for quantum diffusion models with implications for quantum information processing.

Contribution

It proves a sharp phase boundary in quantum reverse diffusion, characterizes the minimum cost of reversal, and establishes an exact Gaussian benchmark for quantum diffusion models.

Findings

Reversal exhibits a noiseless-to-noisy transition at a critical squeezing-to-thermal ratio.

Optimal reverse diffusion minimizes geometric, metrological, and thermodynamic costs simultaneously.

Exact Gaussian reversal of pure quantum states is dynamically unattainable, diverging as 2/t.

Abstract

Classical reverse diffusion is generated by changing the drift at fixed noise. We show that the quantum version of this principle obeys an exact law with a sharp phase boundary. For Gaussian pure-loss dynamics, the canonical model of continuous-variable decoherence, we prove that the unrestricted instantaneous reverse optimum exhibits a noiseless-to-noisy transition: below a critical squeezing-to-thermal ratio, reversal can be noiseless; above it, complete positivity forces irreducible reverse noise whose minimum cost we determine in closed form. The optimal reverse diffusion is uniquely covariance-aligned and simultaneously minimizes the geometric, metrological, and thermodynamic price of reversal. For multimode trajectories, the exact cost is additive in a canonical set of mode-resolved data, and a globally continuous protocol attains this optimum on every mixed-state interval. If a pure nonclassical endpoint is included, the same pointwise law holds for every $t>0$, but the optimum diverges as $2/t$: exact Gaussian reversal of a pure quantum state is dynamically unattainable. These results establish the exact Gaussian benchmark against which any broader theory of quantum reverse diffusion must be measured.