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

In continuously monitored quantum systems, the feedback protocol of Garc'ia-Pintos, Liu, and Gorshkov reshapes the arrow of time: a Hamiltonian $H_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}}=rA\mathrm{/}\tau$ 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, Fr'echet differentiation on the Banach space of trace-class operators, and K"ahler geometry on the pure-state projective manifold, we prove that $\delta \log P_F\mathrm{/}\delta \rho =rA\mathrm{/}\tau =H_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}}$. The Garc'ia-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 $[H,A]\mathrm{\ne }0$), 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.