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Approximating the order 2 quantum Wasserstein distance using the moment-SOS hierarchy

Saroj Prasad Chhatoi, Victor Magron

TL;DR

This work addresses the numerical computation of the order-$2$ quantum Wasserstein distance between density operators by formulating $W_2^2$ as a Generalized Moment Problem on the product of unit spheres and applying a moment-SOS hierarchy to obtain certified, converging lower bounds. The authors establish a GMP representation with marginals matching the input states, prove strong duality at each relaxation level, and prove convergence to the true distance as the relaxation order grows. They likewise connect this to related hierarchies for separability and marginals, and validate the approach through numerical experiments on small quantum systems, including pure-to-pure and mixed-to-mixed cases. The methodology provides a tractable, certificate-based framework for quantum transport distances with potential extensions to broader p-Wasserstein settings and nonlinear polynomial moment problems, offering a path toward scalable quantum OT computations and optimal transport plan extraction.

Abstract

Optimal transport theory has recently been extended to quantum settings, where the density matrices generalize the probability measures. In this paper, we study the computational aspects of the order 2 quantum Wasserstein distance, formulating it as an infinite dimensional linear program in the space of positive Borel measures supported on products of two unit spheres. This formulation is recognized as an instance of the Generalized Moment Problem, which enables us to use the moment-sums of squares hierarchy to provide a sequence of lower bounds converging to the distance. We illustrate our approach with numerical experiments.

Approximating the order 2 quantum Wasserstein distance using the moment-SOS hierarchy

TL;DR

This work addresses the numerical computation of the order- quantum Wasserstein distance between density operators by formulating as a Generalized Moment Problem on the product of unit spheres and applying a moment-SOS hierarchy to obtain certified, converging lower bounds. The authors establish a GMP representation with marginals matching the input states, prove strong duality at each relaxation level, and prove convergence to the true distance as the relaxation order grows. They likewise connect this to related hierarchies for separability and marginals, and validate the approach through numerical experiments on small quantum systems, including pure-to-pure and mixed-to-mixed cases. The methodology provides a tractable, certificate-based framework for quantum transport distances with potential extensions to broader p-Wasserstein settings and nonlinear polynomial moment problems, offering a path toward scalable quantum OT computations and optimal transport plan extraction.

Abstract

Optimal transport theory has recently been extended to quantum settings, where the density matrices generalize the probability measures. In this paper, we study the computational aspects of the order 2 quantum Wasserstein distance, formulating it as an infinite dimensional linear program in the space of positive Borel measures supported on products of two unit spheres. This formulation is recognized as an instance of the Generalized Moment Problem, which enables us to use the moment-sums of squares hierarchy to provide a sequence of lower bounds converging to the distance. We illustrate our approach with numerical experiments.

Paper Structure

This paper contains 15 sections, 5 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Outline and contributions.
  3. Preliminaries and problem statement
  4. Preliminaries on polynomial optimization
  5. Order 2 quantum Wasserstein distance
  6. Main results
  7. An instance of the Generalized Moment Problem
  8. A hierarchy of moment-SOS relaxations
  9. Concluding numerical experiments and perspectives
  10. Example \ref{['ex:pure_pure']} continued.
  11. Example \ref{['ex:mixed_mixed']} continued.
  12. Example maximizing the Wasserstein distance.
  13. Perspectives.
  14. Acknowledgments.
  15. Data availability statement.

Key Result

Lemma 1

Let $\mathbf{X}$ be a compact semialgebraic set and assume that $R - \|\mathbf{x}\|^2 \in \mathcal{Q}(\mathbf{X})_2$. For any $t \in \mathbb{N}$, let assume that $L_t : \mathbb{C}[\mathbf{x},\overline{\mathbf{x}}]_{2t} \to \mathbb{C}$ is a linear functional being nonnegative on $\mathcal{Q}(\mathbf{ Moreover, if then $\{L_t\}_{t \in \mathbb{N}}$ has a point-wise converging subsequence in the spac

Theorems & Definitions (13)

  • Lemma 1
  • Theorem 2
  • Definition 3: beatty2024
  • Example 4: Pure to pure
  • Example 5: Mixed to mixed
  • Definition 6: beatty2024
  • Remark 7
  • Theorem 8
  • proof
  • Theorem 9
  • ...and 3 more