Triangle Inequality for a Quantum Wasserstein Divergence

TL;DR

This paper proves the triangle inequality for a quantum 2-Wasserstein distance, advancing the mathematical foundation of quantum optimal transport using complex analysis techniques.

Contribution

It establishes the triangle inequality for a quantum Wasserstein distance, confirming a conjecture and introducing a new integral representation of the transport cost.

Findings

Proves the triangle inequality for quantum 2-Wasserstein distance.

Introduces a novel integral representation of the optimal transport cost.

Confirms a conjecture by De Palma and Trevisan.

Abstract

We resolve a conjecture of De Palma and Trevisan by proving the triangle inequality for a quantum 2-Wasserstein distance. The proof relies on complex analysis methods to establish a new integral representation of the cost in the optimal transport problem.