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On the metric property of quantum Wasserstein divergences

Gergely Bunth, József Pitrik, Tamás Titkos, Dániel Virosztek

TL;DR

This work investigates whether quantum Wasserstein divergences, as modified quantum OT distances, define genuine metrics on quantum state spaces. It proves a triangle inequality for the quadratic QWDs $d_{\mathcal{A}}$ under the conditions that the intermediate state is pure (or the endpoints are pure) and all states have finite energy, and it provides strong numerical evidence suggesting the inequality holds in general. The authors outline analytic strategies to extend the result to full generality, including qubit-specific lower bounds and operator-mean-based cost estimates, and demonstrate several applications to quantum complexity, mean-field dynamics, and model systems like the Luttinger model. The results establish a solid foundation for using QWDs as metrics in quantum information and dynamics, with potential wide-ranging implications for complexity, transport, and many-body analysis.

Abstract

Quantum Wasserstein divergences are modified versions of quantum Wasserstein distances defined by channels, and they are 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.

On the metric property of quantum Wasserstein divergences

TL;DR

This work investigates whether quantum Wasserstein divergences, as modified quantum OT distances, define genuine metrics on quantum state spaces. It proves a triangle inequality for the quadratic QWDs under the conditions that the intermediate state is pure (or the endpoints are pure) and all states have finite energy, and it provides strong numerical evidence suggesting the inequality holds in general. The authors outline analytic strategies to extend the result to full generality, including qubit-specific lower bounds and operator-mean-based cost estimates, and demonstrate several applications to quantum complexity, mean-field dynamics, and model systems like the Luttinger model. The results establish a solid foundation for using QWDs as metrics in quantum information and dynamics, with potential wide-ranging implications for complexity, transport, and many-body analysis.

Abstract

Quantum Wasserstein divergences are modified versions of quantum Wasserstein distances defined by channels, and they are 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.
Paper Structure (13 sections, 4 theorems, 112 equations, 4 figures)

This paper contains 13 sections, 4 theorems, 112 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Motivation and main result
  3. Basic notions, notation
  4. Triangle inequality for quantum Wasserstein divergences --- the proof of Theorem \ref{['thm:main']}
  5. Numerical evidence for the triangle inequality for generic triplets of states
  6. Steps towards an analytic proof of the triangle inequality in full generality
  7. Triangle inequality for qubits --- the case of the symmetric cost
  8. Lower estimation of the cost function
  9. Applications
  10. Wasserstein complexity
  11. Mean-field approximations of evolution equations
  12. The Luttinger model
  13. Appendix: the code used for the numerical study

Key Result

Theorem 1

Let $\mathcal{H}$ be a separable Hilbert space, and let $\mathcal{A}=\left\{ A_j \right\}_{j=1}^k \subset \mathcal{L}(\mathcal{H})^{sa}$ be an arbitrary finite collection of observable quantities, and let $d_{\mathcal{A}}$ be the corresponding quadratic quantum Wasserstein divergence defined by eq:m holds true.

Figures (4)

  • Figure 1: The plot of the gap in the first scenario: $\mathcal{H}=\mathbb{C}^2,$ the states and the cost are chosen to be nice
  • Figure 2: The plot of the gap in the second scenario: $\mathcal{H}=\mathbb{C}^4,$ the states and the cost are chosen to be nice
  • Figure 3: The plot of the gap in the third scenario: $\mathcal{H}=\mathbb{C}^2,$ the states and the cost are random
  • Figure 4: The plot of the gap in the second scenario: $\mathcal{H}=\mathbb{C}^4,$ the states and the cost are random

Theorems & Definitions (10)

  • Conjecture 1: De Palma-Trevisan, DPT-lecture-notes
  • Theorem 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Corollary 4
  • Remark 1
  • Remark 2