Hybrid Lindblad dynamics and Bayesian inference from stochastic processes
Abstract
We develop a Bayesian formulation of diffusive quantum-classical dynamics by treating the wave function and classical variables as components of an ordinary stochastic process. The joint probability density P(psi,x,t) obeys a classical Fokker-Planck equation, while the quantum state appears as its second moment. Requiring autonomous second-moment evolution and trace preservation, together with a full-column-rank quantum-classical noise correlation matrix, forces the ensemble dynamics into hybrid Lindblad form. The same result also determines the general class of diffusive stochastic unravelings compatible with this hybrid evolution. Notably, this construction makes positivity immediate and does not take complete positivity or an underlying unitary quantum dilation as assumptions, but instead relies on the presence of a sufficiently coupled classical sector. We formulate quantum measurement, filtering and smoothing of hybrid quantum-classical dynamics as standard Bayesian inference and show how to correlation between the quantum and classical noise fully explains the differences between quantum smoothing and smoothing in classical time series models. We recover the stochastic master equation from the Kushner-Stratonovich equation with correlated noise and show how the quantum effect operator is related to the Bayesian backward message through the adjoint dynamics of the linear unraveling. The Bayesian posterior also defines a smoothed density matrix and, more generally, a posterior distribution over latent quantum-classical trajectories. These quantities can be approximated with standard particle filtering and smoothing methods. Numerical examples show that smoothing improves reconstruction of a hidden quantum-classical trajectory and that the full trajectory posterior can retain structure, such as multimodality, that is absent from its density-matrix second moment.
Related
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BibTeX
@misc{kappen2026hybrid,
title = {{Hybrid Lindblad dynamics and Bayesian inference from stochastic processes}},
author = {Hilbert J. Kappen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.21169},
eprint = {2608.21169},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.21169},
}
RIS
TY - UNPB
TI - Hybrid Lindblad dynamics and Bayesian inference from stochastic processes
AU - Hilbert J. Kappen
PY - 2026
DA - 2026/10/03
PB - arXiv
DO - 10.48550/arxiv.2608.21169
UR - https://arxiv.org/abs/2608.21169
AB - We develop a Bayesian formulation of diffusive quantum-classical dynamics by treating the wave function and classical variables as components of an ordinary stochastic process. The joint probability density P(psi,x,t) obeys a classical Fokker-Planck equation, while the quantum state appears as its second moment. Requiring autonomous second-moment evolution and trace preservation, together with a full-column-rank quantum-classical noise correlation matrix, forces the ensemble dynamics into hybrid Lindblad form. The same result also determines the general class of diffusive stochastic unravelings compatible with this hybrid evolution. Notably, this construction makes positivity immediate and does not take complete positivity or an underlying unitary quantum dilation as assumptions, but instead relies on the presence of a sufficiently coupled classical sector. We formulate quantum measurement, filtering and smoothing of hybrid quantum-classical dynamics as standard Bayesian inference and show how to correlation between the quantum and classical noise fully explains the differences between quantum smoothing and smoothing in classical time series models. We recover the stochastic master equation from the Kushner-Stratonovich equation with correlated noise and show how the quantum effect operator is related to the Bayesian backward message through the adjoint dynamics of the linear unraveling. The Bayesian posterior also defines a smoothed density matrix and, more generally, a posterior distribution over latent quantum-classical trajectories. These quantities can be approximated with standard particle filtering and smoothing methods. Numerical examples show that smoothing improves reconstruction of a hidden quantum-classical trajectory and that the full trajectory posterior can retain structure, such as multimodality, that is absent from its density-matrix second moment.
ER -