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arXiv:2501.08066

Wasserstein distances and divergences of order $p$ by quantum channels

Gergely Bunth, József Pitrik, Tamás Titkos, Dániel Virosztek

Published Jan 14, 2025• 0 citations
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Abstract

We introduce a non-quadratic generalization of the quantum mechanical optimal transport problem introduced in [De Palma and Trevisan, Ann. Henri Poincaré, {\bf 22} (2021), 3199-3234] where quantum channels realize the transport. Relying on this general machinery, we introduce $p$-Wasserstein distances and divergences and study their fundamental geometric properties. Finally, we prove triangle inequality for quadratic Wasserstein divergences under the sole assumption that an arbitrary one of the states involved is pure, which is a generalization of our previous result in this direction.

DOI
10.48550/arxiv.2501.08066

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Wasserstein distances and divergences of order ppp by quantum channels
We introduce a non-quadratic generalization of the quantum mechanical optimal transport problem introduced in [De Palma and Trevisan, Ann. Henri Poincaré, {\bf 22} (2021), 3199-3234] where quantum channels realize the transport. Relying on this general machinery, we introduce ppp-Wasserstein distances and divergences and study their fundamental geometric properties. Finally, we prove triangle inequality for quadratic Wasserstein divergences under the sole assumption that an arbitrary one of the states involved is pure, which is a generalization of our previous result in this direction.
Quantum Wasserstein divergences are modified versions of quantum Wasserstein distances defined by channels and they have been conjectured to be genuine metrics on quantum state spaces by De Palma and Trevisan. We prove triangle inequality for quantum Wasserstein divergences for every quantum system described by a separable Hilbert space and any quadratic cost operator under the assumption that a particular state involved is pure and all the states have finite energy. We also provide strong numerical evidence suggesting that the triangle inequality holds in general for an arbitrary choice of states.
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.
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.
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.
We prove Kantorovich duality for a linearized version of a recently proposed non-quadratic quantum optimal transport problem, where quantum channels realize the transport. As an application, we determine optimal solutions of both the primal and the dual problem using this duality in the case of quantum bits and distinguished cost operators, with certain restrictions on the states involved. Finally, keeping the same restrictions regarding the states involved, we use this information on optimal solutions to give an analytical proof of the triangle inequality even for the square of the induced quantum Wasserstein divergences.