The Feedback Hamiltonian is the Score Function: A Diffusion-Model Framework for Quantum Trajectory Reversal

Quantum Physics [Submitted on 23 Apr 2026] The Feedback Hamiltonian is the Score Function: A Diffusion-Model Framework for Quantum Trajectory Reversal Sagar Dubey, Alan John In continuously monitored quantum systems, the feedback protocol of García-Pintos, Liu, and Gorshkov reshapes the arrow of time: a Hamiltonian H_{\mathrm{meas}} = r A / \tau applied with gain X tilts the distribution of measurement trajectories, with X < -2 producing statistically time-reversed outcomes. Why this specific Hamiltonian achieves reversal, and how the mechanism relates to score-based diffusion models in machine learning, has remained unexplained. We compute the functional derivative of the log path probability of the quantum trajectory distribution directly in density-matrix space. Combining Girsanov's theorem applied to the measurement record, Fréchet differentiation on the Banach space of trace-class operators, and Kähler geometry on the pure-state projective manifold, we prove that \delta \log P_F / \delta \rho = r A / \tau = H_{\mathrm{meas}}. The García-Pintos feedback Hamiltonian is the score function of the quantum trajectory distribution -- exactly the object Anderson's reverse-time diffusion theorem requires for trajectory reversal. The identification extends to multi-qubit systems with independent measurement channels, where the score is a sum of local operators. Two consequences follow. First, the feedback gain X generates a continuous one-parameter family of path measures (for feedback-active Hamiltonians with [H, A] \neq 0), with X = -2 recovering the backward process in leading-order linearization -- a structure absent from classical diffusion, where reversal is binary. Second, the score identification enables machine learning (ML) score estimation methods -- denoising score matching, sliced score matching -- to replace the analytic formula when its idealizations (unit efficiency, zero delay, Gaussian noise) fail in real experiments. Comments: 14 pages Subjects: Quantum Physics (quant-ph); Machine Learning (cs.LG) Cite as: arXiv:2604.21210 [quant-ph]   (or arXiv:2604.21210v1 [quant-ph] for this version)   https://doi.org/10.48550/arXiv.2604.21210 Focus to learn more Submission history From: Sagar Dubey [view email] [v1] Thu, 23 Apr 2026 02:02:13 UTC (16 KB) Access Paper: HTML (experimental) view license Current browse context: quant-ph < prev   |   next > new | recent | 2026-04 Change to browse by: cs cs.LG References & Citations INSPIRE HEP NASA ADS Google Scholar Semantic Scholar Export BibTeX Citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Demos Related Papers About arXivLabs Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)