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Quantum Wasserstein distance for Gaussian states

Anaelle Hertz, Mohammad Ahmadpoor, Oleksandr Dzhenzherov, Augusto Gerolin, Khabat Heshami

TL;DR

The paper derives a complete closed-form expression for the quantum Wasserstein distance between any two one-mode Gaussian states within the De Palma–Trevisan transport framework, unifying Gaussian-state transport with classical W2 and thermal-state limits. The authors reduce the problem to optimizing a Gaussian transport map, leverage the Douglas factorization and Schur-positivity constraints, and obtain $D^2( ho_A, ho_B)=\frac{1}{2}\text{Tr}(A+B)-\frac{1}{2}\sqrt{\frac{4\sqrt{\det A}\sqrt{\det B}-2|\sqrt{\det A}-\sqrt{\det B}|-1}{\sqrt{\det A}\sqrt{\det B}}}\,\text{Tr}[\sqrt{\sqrt{B}A\sqrt{B}}]$. This general formula reproduces known thermal-state results, reduces to classical Gaussian $W_2$ in the appropriate limit, and exhibits symmetry, joint convexity, and a bound with the Bures distance, suggesting a robust operational role for Wasserstein distance in quantum state discrimination and metrology. The work also provides concrete examples for thermal and squeezed-thermal states and outlines pathways to multi-mode and non-Gaussian extensions. Overall, it offers a rigorous, tractable bridge between quantum OT and widely used quantum-state distances.

Abstract

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.

Quantum Wasserstein distance for Gaussian states

TL;DR

The paper derives a complete closed-form expression for the quantum Wasserstein distance between any two one-mode Gaussian states within the De Palma–Trevisan transport framework, unifying Gaussian-state transport with classical W2 and thermal-state limits. The authors reduce the problem to optimizing a Gaussian transport map, leverage the Douglas factorization and Schur-positivity constraints, and obtain . This general formula reproduces known thermal-state results, reduces to classical Gaussian in the appropriate limit, and exhibits symmetry, joint convexity, and a bound with the Bures distance, suggesting a robust operational role for Wasserstein distance in quantum state discrimination and metrology. The work also provides concrete examples for thermal and squeezed-thermal states and outlines pathways to multi-mode and non-Gaussian extensions. Overall, it offers a rigorous, tractable bridge between quantum OT and widely used quantum-state distances.

Abstract

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.

Paper Structure

This paper contains 21 sections, 9 theorems, 88 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Definition of quantum optimal transport
  3. Preliminaries for Gaussian states
  4. Phase space formalism and Gaussian states
  5. Gaussian channels
  6. Quantum optimal transport for Gaussian States
  7. Problem Statement
  8. Wasserstein distance between thermal states
  9. Wasserstein distance between Gaussian states related by a unitary transformation
  10. Wasserstein distance between any two Gaussian states
  11. Proof of the main formula
  12. Examples
  13. Properties
  14. Comparison to other distances
  15. Conclusion
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $A$ and $B$ be the covariance matrices of two one-mode Gaussian states $\rho_A$ and $\rho_B$, then the Wasserstein distance is given by

Figures (1)

  • Figure 1: Comparison of distances for two squeezed thermal states $\rho_A$ and $\rho_B$ with covariance matrices $A= (\nu+\theta)\left({\mathrm e}^{-2r}00{\mathrm e}^{2r}\right)$, $B=\nu\left({\mathrm e}^{-2(r+dr)}00{\mathrm e}^{2(r+dr)}\right)$. The labels of the curves refer to the quantities in Table \ref{['tab:comparisonDistancesForThermal']}.

Theorems & Definitions (15)

  • Theorem 1.1
  • proof
  • Lemma C.1
  • Lemma C.2
  • Lemma C.3
  • proof
  • Lemma C.4
  • Lemma C.5
  • proof
  • Lemma C.6
  • ...and 5 more