# Matrix Optimal Mass Transport: A Quantum Mechanical Approach

arXiv: https://arxiv.org/abs/1610.03041 · HTML: https://mlsanity.com/paper/1610.03041/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Oct 2016.

## Built on by (most cited first, 4 of 4)

- [Learning quantum data with the quantum Earth Mover's distance](https://mlsanity.com/paper/2101.03037/) (2021)
- [Riemannian block SPD coupling manifold and its application to optimal transport](https://mlsanity.com/paper/2201.12933/) (2022)
- [Muon Dynamics as a Spectral Wasserstein Flow](https://mlsanity.com/paper/2604.04891/) (2026)
- [Quantum Wasserstein GANs for State Preparation at Unseen Points of a Phase Diagram](https://mlsanity.com/paper/2309.09543/) (2023)

## Abstract (arXiv)

In this paper, we describe a possible generalization of the Wasserstein 2-metric, originally defined on the space of scalar probability densities, to the space of Hermitian matrices with trace one, and to the space of matrix-valued probability densities. Our approach follows a computational fluid dynamical formulation of the Wasserstein-2 metric and utilizes certain results from the quantum mechanics of open systems, in particular the Lindblad equation. It allows determining the gradient flow for the quantum entropy relative to this matricial Wasserstein metric. This may have implications to some key issues in quantum information theory.