Approximating the order 2 quantum Wasserstein distance using the moment-SOS hierarchy
Optimization and Control
2025-11-27 v2
Abstract
Optimal transport theory has recently been extended to quantum settings, where the density matrices generalize the probability measures. In this paper, we study the computational aspects of the order 2 quantum Wasserstein distance, formulating it as an infinite dimensional linear program in the space of positive Borel measures supported on products of two unit spheres. This formulation is recognized as an instance of the Generalized Moment Problem, which enables us to use the moment-sums of squares hierarchy to provide a sequence of lower bounds converging to the distance. We illustrate our approach with numerical experiments.
Cite
@article{arxiv.2506.20006,
title = {Approximating the order 2 quantum Wasserstein distance using the moment-SOS hierarchy},
author = {Saroj Prasad Chhatoi and Victor Magron},
journal= {arXiv preprint arXiv:2506.20006},
year = {2025}
}
Comments
15 pages, this revision includes a dual attainment proof