Exact Stochastic Differential Equations for Quantum Reverse Diffusion

The ensemble-averaged dynamics of open quantum systems are typically irreversible. We show that this irreversibility need not hold at the level of individually monitored quantum trajectories. Our main results are analytical stochastic differential equations for quantum reverse diffusion, along with corresponding stochastic master equations. These equations describe the exact and approximate stochastic reverse processes for continuously monitored Pauli channels, including time-dependent depolarizing noise. We show that the reverse processes generalize the forward dynamics by combining the noise effects of the forward processes with an additional non-Markovian stochastic drift that dynamically steers a quantum state back to its initial configuration. Consequently, the exact SDEs admit closed-form solutions that can be implemented in real-time without the need for variational techniques. Our findings establish an analytical framework for quantum state recovery, noise-resilient quantum gates, quantum generative modelling, quantum tomography via forward-reverse cycles, and potential paradigms for quantum error correction based on reverse diffusion.