Wasserstein Distances and Optimal Transport, A Comprehensive Review for Researchers.

Recent research consolidates developments in Wasserstein distances, a generalisation of optimal transport theory from probability measures to quantum states. This review unifies scattered literature across diverse fields including physics, economics and information theory, identifying open questions and potential future research directions within this evolving area.

The mathematical problem of optimally transporting a distribution of resources, known as optimal transport, underpins diverse fields from economics to statistical physics. Recent attention focuses on extending these tools beyond conventional probability measures to the realm of quantum states, yielding novel distances and divergences with potential applications in areas such as many-body quantum systems and machine learning. However, this expanding body of work remains fragmented, lacking a unified framework for comparison and development. Emily Beatty, affiliated with Université de Lyon, ENS de Lyon, UCBL, CNRS, Inria, and LIP, addresses this need in a comprehensive review titled ‘Wasserstein Distances on Quantum Structures: an Overview’, which synthesises current research and highlights promising directions for future investigation within this evolving field.

Quantum optimal transport distances represent a rapidly developing field that merges classical optimal transport with quantum information theory, expanding both theoretical and applied research opportunities. Researchers investigate Wasserstein-style distances and divergences between quantum states, leveraging established mathematical foundations and exploring applications across diverse scientific disciplines. This dynamic area promises advancements in functional inequalities, many-body physics, machine learning algorithms, and potentially quantum security protocols, driving innovation at the intersection of mathematics, physics, and computer science.

The field currently explores various formulations of quantum Wasserstein distances, seeking a unified and robust measure applicable across different quantum systems and experimental settings. Initial groundwork stems from the work of Zyczkowski and Slomczynski in 1998, who explored the Monge distance between quantum states, laying the conceptual foundation for subsequent investigations. The Monge distance, in classical optimal transport, quantifies the minimum ‘cost’ of transporting a probability distribution into another, and adapting this to quantum states presents significant challenges due to the inherent complexities of quantum mechanics.

A significant focus lies in understanding the relationship between quantum optimal transport and Markovian dynamics, the process by which quantum systems evolve. Wirth demonstrated in 2018 and 2022 how noncommutative transport metrics can describe the evolution of quantum systems as gradients of entropy, providing a valuable tool for understanding complex quantum phenomena. Entropy, in this context, measures the uncertainty or randomness of a quantum state, and its gradient indicates the direction of fastest change. Furthermore, these techniques extend to practical areas like generative adversarial networks (GANs), where Wasserstein distances improve training stability and performance, opening new possibilities for machine learning.

The analysis reveals a strong mathematical foundation, particularly drawing from concepts within noncommutative geometry, a branch of mathematics dealing with spaces where the coordinates do not commute, and highlights the significant role of these distances in diverse applications, ranging from fundamental physics to cutting-edge machine learning.

Current research actively explores the application of quantum optimal transport to areas such as functional inequalities and convergence analysis within many-body physics, seeking to understand the collective behaviour of complex quantum systems and develop new theoretical tools. Many-body physics deals with systems containing a large number of interacting particles, and understanding their collective behaviour is a major challenge in physics.

A central theme emerging from the literature is the ongoing effort to establish a definitive, universally accepted Wasserstein distance for quantum states, a challenge that continues to drive research and inspire new theoretical developments. Several works focus on defining and analysing specific quantum distance measures, each offering unique properties and potential applications, allowing researchers to tailor their approach to specific problems and experimental settings.

This proliferation of measures underscores the complexity of adapting classical optimal transport to the quantum realm and the need for careful consideration of their respective strengths and limitations. A notable trend within the examined literature is the increasing adoption of open-access publishing platforms, such as the journal Quantum, suggesting a commitment to wider dissemination of research findings and fostering collaborative efforts within the scientific community. Future research directions, as identified within the reviewed works, include addressing open problems related to the definition of optimal distances and exploring novel applications in areas such as quantum machine learning and security protocols.

The potential connection to emerging technologies like deepfake detection suggests a growing practical relevance for these theoretical developments, highlighting the potential for quantum optimal transport to address real-world challenges. Continued investigation into the interplay between quantum optimal transport and other areas of physics and information theory promises to yield further insights and advancements, solidifying its position as a vibrant and impactful field of research. Researchers actively consolidate a growing body of work concerning Wasserstein-style distances and divergences between quantum states, building upon the established theory of optimal transport and identifying a fragmented literature.