Generating quantum ensembles via reverse-time quantum diffusions

We establish a reverse-time denoising theory for quantum diffusions of continuously measured quantum systems. Starting from the stochastic Schr\"odinger equation of a forward noising dynamics, we derive the exact reverse-time dynamics for quantum trajectories, whose law coincides with the time-reversal of the original process. We prove that the denoising dynamics is a physically admissible quantum diffusion, with the same measurement-induced noise but a state-dependent feedback Hamiltonian, a direct analogue of the "score function" of generative classical diffusion models. This provides a principled framework for converting samples of a simple distribution into those of a more complex ensemble of quantum states. We show how the denoising dynamics can be directly learnt from forward trajectory data, and how to exploit purification to initialise the denoising process.