发表机构
Donders Institute, Radboud University（道德斯研究所，拉德堡德大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究建立了混合量子-经典动力学的贝叶斯公式化，将量子-经典状态估计转化为经典隐态推断问题，利用粒子滤波等方法近似相关量，平滑可改善隐轨迹重建，框架关联了量子滤波等与贝叶斯时间序列推断机制。
AI 中文摘要
我们通过将波函数和经典变量视为普通随机过程的组成部分，建立了扩散型量子-经典动力学的贝叶斯公式化。联合概率密度P(ψ,x,t)服从经典的福克-普朗克方程，而量子态表现为其二阶矩。要求该二阶矩线性且自主演化，可得到混合林德布拉德方程及其随机展开。该构造使正定性和展开自由变得直接，并通过协方差矩阵(C,Γ,Q)统一描述量子噪声、经典噪声及其相关性。同一随机表示将量子-经典状态估计转化为经典隐态推断问题，滤波和平滑是对观测到的经典轨迹进行贝叶斯条件化。我们从含相关噪声的库什纳-斯特拉托诺维奇方程中恢复随机主方程，并展示量子效应算子如何通过线性展开伴随动力学与贝叶斯反向消息关联。贝叶斯后验还定义了平滑密度矩阵，更一般地，定义了隐量子-经典轨迹的后验分布，这些量可通过标准粒子滤波和平滑方法近似。数值例子表明，平滑可改善隐量子-经典轨迹的重建，且完整轨迹后验可保留其密度矩阵二阶矩中不存在的结构（如多模态）。所得框架将量子滤波、回溯和平滑与贝叶斯时间序列推断的标准前向-后向机制关联起来。
英文摘要
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.
Comments27 pages, 3 figures, 1 table
