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

Gergely Bunth, József Pitrik, Tamás Titkos, Dániel Virosztek

TL;DR

The paper develops a non-quadratic quantum OT framework in which transport is realized by quantum channels and introduces $p$-Wasserstein distances $D_{A,p}$ and divergences $d_{A,p}$ via cost operators $C_{A,p}$. It establishes existence of optimal transport plans under a finite-energy condition and analyzes how cost-operator sets vary with $p$, revealing monotonicity and special-qubit equalities. The authors prove a triangle inequality for the quadratic divergences $d_{A,2}$ under the sole assumption that one of the states is pure, while showing that for $p<2$ the triangle inequality can fail in general. These results connect non-quadratic quantum OT to the quadratic DPT framework, provide a structured view of cost operators across $p$, and advance the understanding of geometry on quantum state spaces under transport-like costs.

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.

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

TL;DR

The paper develops a non-quadratic quantum OT framework in which transport is realized by quantum channels and introduces -Wasserstein distances and divergences via cost operators . It establishes existence of optimal transport plans under a finite-energy condition and analyzes how cost-operator sets vary with , revealing monotonicity and special-qubit equalities. The authors prove a triangle inequality for the quadratic divergences under the sole assumption that one of the states is pure, while showing that for the triangle inequality can fail in general. These results connect non-quadratic quantum OT to the quadratic DPT framework, provide a structured view of cost operators across , and advance the understanding of geometry on quantum state spaces under transport-like costs.

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 -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.
Paper Structure (12 sections, 16 theorems, 198 equations)

This paper contains 12 sections, 16 theorems, 198 equations.

Table of Contents

  1. Introduction
  2. Motivation and main result
  3. Basic notions, notation
  4. Quantum mechanical analogues of the general non-quadratic optimal transport problem on Euclidean spaces
  5. A quantum mechanical optimal transport problem generalizing DPT-AHP
  6. Important special cases.
  7. A quantum mechanical optimal transport problem without imposing independence
  8. $p$-Wasserstein distances and divergences
  9. Relations between the sets of all possible cost operators for different values of $p$
  10. Triangle inequality is inherited by larger parameter values
  11. $p=2$ as a possible threshold for the triangle inequality
  12. Triangle inequality for quantum Wasserstein divergences --- the proof of Theorem \ref{['thm:main']}

Key Result

Proposition 1

Let $c: \mathbb{R}^K \times \mathbb{R}^K \rightarrow \mathbb{R}$ be a non-negative and lower semi-continuous function, let $\mathcal{A}$ be a finite collection of observables on $\mathcal{H},$ and $\rho, \omega \in \mathcal{S}\left( \mathcal{H} \right)$ given marginals. Then there exists an optimal

Theorems & Definitions (33)

  • Proposition 1: Existence of optimal plans
  • proof
  • Definition 1
  • Remark 2
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • Corollary 5
  • Lemma 6
  • ...and 23 more