从非线性随机微分方程到量子信道：柯尔莫哥洛夫-林德布拉德映射
From Nonlinear Stochastic Differential Equations to Quantum Channels: The Kolmogorov--Lindblad Mapping
- The Pennsylvania State University（宾夕法尼亚州立大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
研究针对非线性SDE难以直接量子表示的问题，提出柯尔莫哥洛夫-林德布拉德映射（KLM），将经典SDE转化为量子信道，通过数值实验验证其可观测量收敛性，为经典随机动力学的量子计算提供了可控路径。
AI中文摘要：
非线性随机微分方程（SDE）是分子建模、药物发现、量化金融、随机学习和不确定性量化的基础。因此，它们的期望、事件概率和时间相关性自然成为量子计算的目标，但非线性漂移和噪声实现的平均阻碍了直接量子表示。我们在概率定律层面开发了精确的柯尔莫哥洛夫-林德布拉德映射（KLM），其被天然编码为迹为1的量子密度算子。对于每个布朗运动实现，路径密度允许一个半密度，其演化遵循随机薛定谔方程。对相关纯态取平均得到Γ(t)的林德布拉德方程，其对角核恰好是福克-普朗克密度p(t,x)=Γ(t;x,x)，经典扩散通过厄米跳算子的退相干表示。统计可观测量成为量子期望，且不存在直接对密度进行振幅编码引入的未知时变归一化，经典双时间相关性允许精确的量子回归公式。保结构的伽辽金投影在有限维下保留林德布拉德形式，将经典SDE和开放量子动力学置于相同的量子原生计算基础上。双阱朗之万动力学和含噪洛伦兹-63的数值实验显示统计可观测量快速收敛。因此，KLM提供了从一般非线性随机动力学到量子信道的数学可控路径，同时将函数逼近和相干算子访问作为算法效率的剩余决定因素。
英文摘要：
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.
