Quantum Optimal Transport: Quantum Channels and Qubits

TL;DR

This paper explores two recent approaches to quantum optimal transport, involving quantum channels and Hamming-Wasserstein distances on qubits, providing an elementary overview in finite-dimensional quantum systems.

Contribution

It introduces and explains two novel methods for quantum optimal transport, focusing on quantum channels and Hamming-Wasserstein distances, in an accessible manner.

Findings

Introduction of quantum channels as transport plans

Definition of Hamming-Wasserstein distance for qubits

Elementary presentation in finite-dimensional quantum systems

Abstract

These notes are based on the lectures given by the second author at the School on Optimal Transport on Quantum Structures at Erd\"os Center in September 2022. The focus of the exposition is on two recently introduced approaches on quantum optimal transport: one based on quantum channels as generalized transport plans, the other based on the notion of Hamming-Wasserstein distance of order 1 on multiple-qubit systems. The material is presented in an elementary manner with a focus on the finite-dimensional setting.