# Score Reversal Is Not Free for Quantum Diffusion Models

arXiv: https://arxiv.org/abs/2603.06488 · HTML: https://mlsanity.com/paper/2603.06488/

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- First posted on arXiv 6 Mar 2026.

## Abstract (arXiv)

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.