Triangle Inequality for a Quantum Wasserstein Divergence

2025/11/25 by Wirth, Melchior · 2 citations
#FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Mathematical Physics (math-ph) #Quantum Physics (quant-ph)

paper · doi:10.48550/arxiv.2511.20450

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.

Citations

• Strong Kantorovich duality for quantum optimal transport with generic cost and optimal couplings on quantum bits
• Wasserstein Distances on Quantum Structures: an Overview
• Wasserstein distances and divergences of order p by quantum channels
• Towards Optimal Transport for Quantum Densities
• A Noncommutative Transport Metric and Symmetric Quantum Markov Semigroups as Gradient Flows of the Entropy
• L²-Wasserstein distances of tracial W^*-algebras and their disintegration problem
• Matrix Optimal Mass Transport: A Quantum Mechanical Approach
• Gradient flow and entropy inequalities for quantum Markov semigroups\n with detailed balance
• Gradient flow and entropy inequalities for quantum Markov semigroups with detailed balance
• Matrix-valued Monge-Kantorovich Optimal Mass Transport
• A Free Probability Analogue of the Wasserstein Metric on the Trace-State Space

Cited by

• The swap transpose on couplings translates to Petz' recovery map on quantum channels
• Quantum Wasserstein isometries of the n-qubit state space: a Wigner-type result
• Quantum Wasserstein distance and its relation to several types of fidelities

Related

• Wasserstein distances and divergences of order p by quantum channels
• Quantum Optimal Transport: Quantum Couplings and Many-Body Problems
• A Duality-Based Proof of the Triangle Inequality for the Wasserstein Distances
• Quantum Wasserstein distance for Gaussian states
• Wasserstein Distances on Quantum Structures: an Overview