From Nonlinear Stochastic Differential Equations to Quantum Channels: The Kolmogorov--Lindblad Mapping
Abstract
Nonlinear stochastic differential equations describe dynamics under uncertainty, but their nonlinear coefficients and noise averaging complicate quantum representations. We develop a Kolmogorov--Lindblad mapping that encodes their probability laws as the position diagonals of trace-one quantum density operators. For each Brownian realisation, a stochastic flow transports the initial ensemble; the square-root Jacobian makes its action on half-densities unitary. Averaging the resulting pure-state projectors gives a Lindblad equation with Hermitian jump operators. Its diagonal reproduces the Fokker--Planck density, while forward and backward intertwining identities recover bounded observables and time correlations independently of the initial coherences. A Galerkin approximation obtained by projecting the Stratonovich generators preserves the Lindblad structure. We give a residual-based error estimate and conditional quantum costs that display the dimension dependence of approximation constants, operator normalisations, state preparation and readout. Numerical experiments for double-well Langevin dynamics and noisy Lorenz--63 show rapid convergence of selected statistics at fixed dimension. The construction provides an exact bridge from flow-regular nonlinear diffusions to quantum channels; any computational advantage additionally requires controlled approximation and coherent access for the chosen problem family.
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BibTeX
@misc{wu2026from,
title = {{From Nonlinear Stochastic Differential Equations to Quantum Channels: The Kolmogorov--Lindblad Mapping}},
author = {Hsuan-Cheng Wu and Xiantao Li},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.09903},
eprint = {2608.09903},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.09903},
}
RIS
TY - UNPB
TI - From Nonlinear Stochastic Differential Equations to Quantum Channels: The Kolmogorov--Lindblad Mapping
AU - Hsuan-Cheng Wu
AU - Xiantao Li
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2608.09903
UR - https://arxiv.org/abs/2608.09903
AB - Nonlinear stochastic differential equations describe dynamics under uncertainty, but their nonlinear coefficients and noise averaging complicate quantum representations. We develop a Kolmogorov--Lindblad mapping that encodes their probability laws as the position diagonals of trace-one quantum density operators. For each Brownian realisation, a stochastic flow transports the initial ensemble; the square-root Jacobian makes its action on half-densities unitary. Averaging the resulting pure-state projectors gives a Lindblad equation with Hermitian jump operators. Its diagonal reproduces the Fokker--Planck density, while forward and backward intertwining identities recover bounded observables and time correlations independently of the initial coherences. A Galerkin approximation obtained by projecting the Stratonovich generators preserves the Lindblad structure. We give a residual-based error estimate and conditional quantum costs that display the dimension dependence of approximation constants, operator normalisations, state preparation and readout. Numerical experiments for double-well Langevin dynamics and noisy Lorenz--63 show rapid convergence of selected statistics at fixed dimension. The construction provides an exact bridge from flow-regular nonlinear diffusions to quantum channels; any computational advantage additionally requires controlled approximation and coherent access for the chosen problem family.
ER -