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.