AI 中文总结
该研究建立了海森堡表象与贝叶斯逆问题的形式对应，通过理论推导得出四个核心定理，并结合数值结果验证了结论的合理性。
AI 中文摘要
该研究提出的公式通过将观测噪声算子与初始可观测量等同，建立了海森堡表象与逆问题贝叶斯公式之间的形式对应关系。初始可观测量与希尔伯特空间中的二次似然范数相关联，该范数依赖于含噪观测值和由未知参数参数化的直接模型。利用Karhunen-Loève展开，量子态以Matern协方差的特征值和特征函数表示。随后，似然概率以观测态和依赖于模型的态表示，并被重写为量子态的二次范数，其中观测噪声协方差被提议与初始可观测量等同。主要结果由四个定理总结：演化算子的幺正性、哈密顿量的自伴性、可观测量的时间不变性，以及海森堡演化与贝叶斯后验概率之间的等价性。当长度参数趋于0和无穷大时，分析了Matern协方差的极限情况。最后，数值结果支持了这些分析结论。
英文摘要
The proposed formulation establishes a formal correspondence between the Heisenberg representation and the Bayesian formulation of inverse problems. The likelihood function is first expressed in Hilbert space through a quadratic norm of the difference between the noisy observations and the response of a forward model depending on the unknown parameters, weighted by the inverse of the observational noise covariance. Based on the common mathematical structure of this quadratic form and the observable operator in the Heisenberg representation, the inverse observational noise covariance operator is identified with the initial observable. To establish this correspondence, a stochastic field with Matérn covariance is considered and represented through the Karhunen-Loève expansion. The eigenvalues and eigenfunctions of this expansion are then used to define the state $|Ψ\rangle$ within the proposed formalism. The main results are summarized by four theorems: the unitarity of the evolution operator, the self-adjointness of the Hamiltonian, the conservation of the trace of the observable, and the formal correspondence between the Bayesian likelihood and the Heisenberg observable structure. Limiting cases of the Matérn covariance are analyzed. Finally, numerical examples illustrate the analytical results.
CommentsRevised version 2: corrections to the mathematical formulation and proofs, with improvements in notation and presentation
