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Quantum Diffusion Models: Score Reversal Is Not Free in Gaussian Dynamics

Ammar Fayad

Abstract

Diffusion-based generative modeling suggests reversing a noising semigroup by adding a score drift. For continuous-variable Gaussian Markov dynamics, complete positivity couples drift and diffusion at the generator level. For a quantum-limited attenuator with thermal parameter $ν$ and squeezing $r$, the fixed-diffusion Wigner-score (Bayes) reverse drift violates CP iff $\cosh(2r)>ν$. Any Gaussian CP repair must inject extra diffusion, implying $-2\ln F\ge c_{\text{geom}}(ν_{\min})I_{\mathrm{dec}}^{\mathrm{wc}}$.

Quantum Diffusion Models: Score Reversal Is Not Free in Gaussian Dynamics

Abstract

Diffusion-based generative modeling suggests reversing a noising semigroup by adding a score drift. For continuous-variable Gaussian Markov dynamics, complete positivity couples drift and diffusion at the generator level. For a quantum-limited attenuator with thermal parameter and squeezing , the fixed-diffusion Wigner-score (Bayes) reverse drift violates CP iff . Any Gaussian CP repair must inject extra diffusion, implying .
Paper Structure (15 sections, 4 theorems, 42 equations, 2 figures)

This paper contains 15 sections, 4 theorems, 42 equations, 2 figures.

Table of Contents

  1. Introduction.
  2. Gaussian quantum channels and complete positivity.
  3. Classical score reversal is free; quantum is not.
  4. No-go theorem.
  5. Physical picture.
  6. The quantum noise floor.
  7. Numerical validation.
  8. Discussion.
  9. Conventions and objects used in the Letter
  10. From HHW CP to the generator CP matrix
  11. CP violation witnessed by an unphysical TMSV output (Choi/TMSV test)
  12. Reverse-time sign conventions and CP-spectrum invariance under K->-K
  13. Petz metric comparison and the constant cgeom(nu)
  14. Endpoint fidelity bound
  15. Minimal CP repair as a convex SDP

Key Result

Theorem 1

The Bayes reverse candidate eq:revdrift has generator CP matrix Let $u = \tfrac{1}{\sqrt{2}}(1,i)^T$ (so $\sigma u = iu$). Then Consequently, $M^{\mathrm{Bayes}}\not\succeq 0$ (CP violation) iff In particular, for unsqueezed thermal references ($r=0$) the obstruction never activates for $\nu\ge 1$; squeezing is the source of CP failure in this phase-covariant one-mode setting.

Figures (2)

  • Figure 1: No-go phase diagram for the fixed-diffusion score-lift (HHW conventions, vacuum $\nu=1$). Left: minimum eigenvalue $\lambda_{\min}(M^{\mathrm{Bayes}})$ over $(\nu,r)$; CP violation occurs exactly where $\lambda_{\min}<0$. The analytic threshold curve $\cosh(2r)=\nu$ is overlaid. Right: magnitude $\mathrm{Tr}(\Delta D_{\mathrm{qu}}^\star)$ of the minimal CP repair diffusion (SDP), showing mandatory noise injection beyond the CP boundary.
  • Figure 2: Quantum noise floor for Gaussian decoders. Worst-case infidelity $-2\ln F$ (solid) and the lower bound $c_{\mathrm{geom}}(\nu_{\min})\,\mathcal{I}_{\mathrm{dec}}^{\mathrm{wc}}(S)$ (dashed) versus depth $S$ for three representative HHW settings $(\nu,r)$. The shaded band marks depths where $\lambda_{\min}(M^{\mathrm{Bayes}})<0$ (CP defect, repair required), and the inset zooms the theorem lower bound. Details are in Supplemental Material SM.

Theorems & Definitions (6)

  • Theorem 1: No-go for noiseless quantum score reversal
  • Theorem 2: Quantum noise floor for Gaussian decoders
  • Lemma 1: Spectrum invariance under drift sign flip
  • proof
  • Lemma 2: Adjacent-ratio restriction for displacement tangents
  • proof