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arXiv:2308.11134

Quantum Optimal Transport: Quantum Couplings and Many-Body Problems

François Golse

Published Aug 22, 2023• 0 citations
arXivHTMLPDF

Abstract

This text is a set of lecture notes for a 4.5-hour course given at the Erdös Center (Rényi Institute, Budapest) during the Summer School "Optimal Transport on Quantum Structures" (September 19th-23rd, 2023). Lecture I introduces the quantum analogue of the Wasserstein distance of exponent $2$ defined in [F. Golse, C. Mouhot, T. Paul: Comm. Math. Phys. 343 (2016), 165-205], and in [F. Golse, T. Paul: Arch. Ration. Mech. Anal. 223 (2017) 57-94]. Lecture II discusses various applications of this quantum analogue of the Wasserstein distance of exponent $2$, while Lecture III discusses several of its most important properties, such as the triangle inequality, and the Kantorovich duality in the quantum setting, together with some of their implications.

DOI
10.48550/arxiv.2308.11134

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This text is a set of lecture notes for a 4.5-hour course given at the Erdös Center (Rényi Institute, Budapest) during the Summer School "Optimal Transport on Quantum Structures" (September 19th-23rd, 2023). Lecture I introduces the quantum analogue of the Wasserstein distance of exponent 222 defined in [F. Golse, C. Mouhot, T. Paul: Comm. Math. Phys. 343 (2016), 165-205], and in [F. Golse, T. Paul: Arch. Ration. Mech. Anal. 223 (2017) 57-94]. Lecture II discusses various applications of this quantum analogue of the Wasserstein distance of exponent 222, while Lecture III discusses several of its most important properties, such as the triangle inequality, and the Kantorovich duality in the quantum setting, together with some of their implications.
An analogue of the quadratic Wasserstein (or Monge-Kantorovich) distance between Borel probability measures on Rd\mathbf{R}^dRd has been defined in [F. Golse, C. Mouhot, T. Paul: Commun. Math. Phys. 343 (2015), 165-205] for density operators on L2(Rd)L^2(\mathbf{R}^d)L2(Rd), and used to estimate the convergence rate of various asymptotic theories in the context of quantum mechanics. The present work proves a Kantorovich type duality theorem for this quantum variant of the Monge-Kantorovich or Wasserstein distance, and discusses the structure of optimal quantum couplings. Specifically, we prove that optimal quantum couplings involve a gradient type structure similar to the Brenier transport map (which is the gradient of a convex function), or more generally, to the subdifferential of a l.s.c. convex function as in the Knott-Smith optimality criterion (see Theorem 2.12 in [C. Villani: "Topics in Optimal Transportation", Amer. Math. Soc. 2003]).
Optimal transport between classical probability distributions has been proven useful in areas such as machine learning and random combinatorial optimization. Quantum optimal transport, and the quantum Wasserstein distance as the minimal cost associated with transforming one quantum state to another, is expected to have implications in quantum state discrimination and quantum metrology. In this work, following the formalism introduced in [De Palma, G. and Trevisan, D. Ann. Henri Poincaré, {\bf 22} (2021), 3199-3234] to compute the optimal transport plan between two quantum states, we give a general formula for the Wasserstein distance of order 2 between any two one-mode Gaussian states. We discuss how the Wasserstein distance between classical Gaussian distributions and the quantum Wasserstein distance by De Palma and Trevisan for thermal states can be recovered from our general formula for Gaussian states. This opens the path to directly compare various known distance measures with the Wasserstein distance through their closed-form solutions.
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 generalization 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 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.
Friedland et al. [PRL 129, 110402 (2022)] proposed and studied a quantum analogue of the ppp-Wasserstein distance based on quantum cost matrices and quantum couplings. They conjectured that, despite being only a semidistance in general, this quantity is a true distance for a particular quantum cost matrix and for cost matrices in a small neighborhood of it. We disprove these conjectures by exhibiting an explicit family of triples of states for which the triangle inequality fails.
We prove a new version of Egorov's theorem formulated in the Schrödinger picture of quantum mechanics, using the ppp-Wasserstein metric applied to the Husimi functions of quantum states. The special case p=1p=1p=1 corresponds to a "low-regularity" Egorov theorem, while larger values p>1p>1p>1 yield progressively stronger estimates. As a byproduct of our analysis, we prove an optimal transport inequality analogous to a result of Golse and Paul in the context of mean-field many-body quantum mechanics.
We develop a Quantum Optimal Transport (QOT) barycenter framework for quantum states, as an analog of Wasserstein barycenters~\cite{AguCar}, and establish existence and duality results for a broad class of possibly unbounded transport costs on separable Hilbert spaces. Our framework provides, in particular, a unified treatment of 222-quantum Wasserstein (QW) barycenters in both the quantum-state and quantum-channel formulations by specializing to the canonical quadratic cost operators associated with the 222-quantum Wasserstein distances of Caglioti--Golse--Mouhot--Paul~\cite{caglioti2021towards,Golse-Mouhot-Paul:2016} and De Palma--Trevisan~\cite{DPaTre19}. The central results concern Gaussian input states and, in particular, Gaussian rigidity: whether Gaussian input states force the 2-QW barycenter itself to be Gaussian and uniquely determined when the minimization is taken over all quantum states. We first show that the barycenter problem admits a Gaussian minimizer and reduces to a finite-dimensional convex optimization problem over covariance matrices. The main difficulty is that uniqueness of the optimal covariance does not, in general, imply uniqueness of the underlying quantum state. We bridge this gap through a state-reconstruction principle under covariance complementary slackness that upgrades uniqueness of the optimal covariance to uniqueness of the full quantum state, and thereby prove a global rigidity theorem: if at least one Gaussian input is faithful, then the barycenter is unique among all quantum states and is necessarily Gaussian. Faithfulness is sufficient but not necessary: families of pure inputs still determine a unique barycenter, whereas partially pure nonfaithful Gaussian inputs may admit multiple barycenters.