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

Abstract

In continuously monitored quantum systems, the feedback protocol of Garc\'ia-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, Fr\'echet differentiation on the Banach space of trace-class operators, and K\"ahler geometry on the pure-state projective manifold, we prove that $\delta \log P_F / \delta \rho = r A / \tau = H_{\mathrm{meas}}$. 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.