Optimal control theory for measured quantum Schrödinger bridges
Abstract
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.
Related
- Quantum-Enhanced Sampling of Schrödinger Bridges
- Bridge of Ψ's: Quantum Circuit Optimization with Schrödinger Bridges
- Random layers for quantum optimal control with exponential expressivity
- Quantum Optimal Control Primer
- Model predictive quantum control: a modular approach for efficient and robust quantum optimal control
BibTeX
@misc{ohzeki2026optimal,
title = {{Optimal control theory for measured quantum Schrödinger bridges}},
author = {Masayuki Ohzeki and Andrew N. Jordan},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.03097},
eprint = {2609.03097},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.03097},
}
RIS
TY - UNPB
TI - Optimal control theory for measured quantum Schrödinger bridges
AU - Masayuki Ohzeki
AU - Andrew N. Jordan
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.03097
UR - https://arxiv.org/abs/2609.03097
AB - 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.
ER -