Optimal control theory for measured quantum Schrödinger bridges

How can the mathematical framework of Schrödinger bridges—used to interpolate between probability distributions—be applied to the control of continuously monitored quantum systems? Specifically, the authors investigate whether the computational potentials (Sinkhorn/Doob potentials) used to solve these bridge problems have a direct physical interpretation in the context of quantum measurement and control.

The authors analyze the diffusion induced on the manifold of quantum states by continuous measurement. By conditioning this diffusion on terminal distributions or measurement effects, they derive a Fokker-Planck equation on quantum state space. They then show that the backward bridge potential, which governs the optimal drift toward a target, can be linked to the Chantasri-Dressel-Jordan (CDJ) path-integral framework. By applying the Dressel-Jordan response formula, they relate the logarithmic directional derivative of these bridge potentials to the imaginary part of generalized weak values.

The study establishes that the bridge potential is not merely an abstract mathematical tool but a physical backward likelihood of a terminal event. Its response to a Hamiltonian perturbation is exactly twice the imaginary part of the corresponding generalized weak value. This provides a direct feedback rule: the imaginary weak value acts as a control score that dictates how to steer the system toward a desired endpoint. The authors illustrate this with three examples: an effect-conditioned collapse, a full terminal-distribution bridge, and a two-detector setup measuring noncommuting observables. In the latter, they demonstrate that using the weak-value/Sinkhorn-score drift significantly reduces the L1 distance between the actual and target terminal distributions compared to uncontrolled processes.

This work bridges the gap between optimal transport theory and quantum control. It provides a systematic pipeline for quantum control synthesis: discretize the state-space kernel, compute the Sinkhorn potential, convert it into local weak-value scores, and project these onto available control directions. This approach identifies weak measurement as a natural mechanism for entropic regularization, offering a robust route from theoretical Sinkhorn scaling to practical Hamiltonian control synthesis.