Wasserstein Distances on Quantum Structures: an Overview

TL;DR

This paper reviews the development of quantum Wasserstein distances, highlighting their applications, current state, open problems, and future directions in quantum optimal transport.

Contribution

It consolidates scattered research on quantum Wasserstein distances, providing a comprehensive overview and identifying open problems and future research avenues.

Findings

Quantum Wasserstein distances have advanced functional inequalities and convergence analysis.

Applications include quantum generative models and many-body physics.

The literature is fragmented with no consensus on a 'true' quantum Wasserstein distance.

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.