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 Hmeas = r A / τ 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 δ log PF / δ ρ = r A / τ = Hmeas. 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] 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.