Wasserstein Distances on Quantum Structures: an Overview
Emily Beatty
TL;DR
This survey provides a structured overview of quantum generalisations of Wasserstein distances, organizing approaches into coupling, dynamical, and Lipschitz frameworks. It details foundational classical OT concepts, translates them into quantum language via density operators and von Neumann algebras, and surveys major constructions such as $W_2^{MK,\epsilon}$, $W_2^S$, modular transport plans, and $W_1^H$, among others. The article catalogs key properties (faithfulness, triangle inequality, data processing, duality) and highlights obstacles arising from noncommutativity and the quantum marginal problem, explaining why a single 'true' quantum Wasserstein distance remains elusive. It also reviews dynamical gradient-flow perspectives, Ricci curvature bounds for quantum Markov semigroups, and numerous applications to Schrödinger-type limits, quantum Gaussian systems, and quantum GANs, while outlining open directions for linking frameworks, achieving quantitative equivalences, and advancing numerical methods. Collectively, the work maps the landscape of quantum OT, clarifies trade-offs, and points to practical implications for quantum information, computation, and learning through Wasserstein-type tools.
Abstract
The theory of optimal transport of probability measures has wide-ranging applications across a number of different fields, including concentration of measure, machine learning, Markov chains, and economics. The generalisation of optimal transport tools from probability measures to quantum states has shown great promise over the last few years, particularly in the development of the theory of Wasserstein-style distances and divergences between quantum states. Such distances have already led to a broad range of developments in the quantum setting such as functional inequalities, convergence of solutions in many-body physics, improvements to quantum generative adversarial networks, and more. However, the literature in this field is quite scattered, with very few links between different works and no real consensus on a `true' quantum Wasserstein distance. The aim of this review is to bring these works together under one roof and give a full overview of the state of the art in the development of quantum Wasserstein distances. We also present a variety of open problems and unexplored avenues in the field, and examine the future directions of this promising line of research. This review is written for those interested in quantum optimal transport in coming from both the fields of classical optimal transport and of quantum information theory, and as a resource for those working in one area of quantum optimal transport interested in how existing work may relate to their own.