发表机构
Graduate School of Information Sciences, Tohoku University; Department of Physics, Institute of Science Tokyo; Research and Education Institute for Semiconductors and Informatics, Kumamoto University; Sigma-i Co., Ltd.; Institute for Quantum Studies, Chapman University; The Kennedy Chair in Physics, Chapman University; Department of Physics and Astronomy, University of Rochester（东北大学情报科学研究院; 东京科学大学物理学系; 熊本大学半导体与信息学研究教育院; Sigma-i有限公司; 查普曼大学量子研究学院; 查普曼大学肯尼迪物理学讲席; 罗切斯特大学物理与天文系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究将薛定谔桥与最优控制理论结合，推导连续监测量子系统的控制解，建立其与最优路径框架的关联，为量子控制提供新方法。
AI 中文摘要
薛定谔桥与熵最优传输通常被表述为初始概率分布与最终概率分布之间的随机插值问题。在其计算形式中，桥势通过Sinkhorn算法或迭代比例拟合得到，常被视为辅助缩放函数。我们证明，对于连续监测的量子系统，这些势具有直接的测量理论意义。条件量子轨迹理论的数学结构在量子态空间上诱导出Fokker-Planck方程。将该扩散过程以终端分布或终端测量效应为条件，会产生Doob/Sinkhorn势，其沿幺正控制向量场的方向导数为广义弱值的虚部。同一构造将薛定谔桥视角与连续监测轨迹的最优路径框架关联起来：后向桥势扮演类效应协态的角色，其弱值方向导数给出局部控制信号。通过指定期望端点分布，薛定谔桥解所诱导的漂移为最小化二次成本反馈律的控制解，引导分布至期望端点。我们将可用控制哈密顿量的控制得分量化为桥势的对数方向导数或虚弱值。三个显式例子表明，弱测量是量子态传输的自然熵正则化机制，并提供了从Sinkhorn缩放到量子反馈及实际最优控制的哈密顿量控制综合的途径。
英文摘要
Schrödinger bridges and entropic optimal transport are usually formulated as stochastic interpolation problems between initial and final probability distributions. In their computational form, the bridge potentials are obtained by Sinkhorn or iterative proportional fitting, and are often regarded as auxiliary scaling functions. We show that for continuously monitored quantum systems, these potentials acquire a direct measurement-theoretic meaning. The mathematical structure of conditional quantum trajectory theory induces a Fokker--Planck equation on quantum state space. Conditioning this diffusion on a terminal distribution or a terminal measurement effect produces a Doob/Sinkhorn potential whose directional derivative along a unitary control vector field is the imaginary part of a generalized weak value. The same construction connects the Schrödinger-bridge viewpoint to the optimal-path framework for continuously monitored trajectories: the backward bridge potential plays the role of an effect-like costate, and its weak-value directional derivative gives the local control signal. By specifying the desired endpoint distribution, the induced drift produced by the Schrödinger bridge solution is the control solution that minimizes the quadratic cost feedback law, guiding the distribution to its desired endpoint. We quantify the control score of the available control Hamiltonian as a logarithmic directional derivative of the bridge potential, or an imaginary weak value. Three explicit examples identify weak measurement as a natural entropic regularization mechanism for quantum state transport and gives a route from Sinkhorn scaling to quantum feedback and Hamiltonian control synthesis for practical optimal control.
Comments13 pages, 3 figures
