Quantum Optimal Transport: Quantum Channels and Qubits
Giacomo De Palma, Dario Trevisan
TL;DR
This work surveys two modern approaches to quantum optimal transport in finite dimensions: (i) using quantum channels as transport plans to map one state to another, guided by purification-based couplings, and (ii) a quantum Wasserstein-1 distance for qubits inspired by the classical Hamming distance. It develops rigorous foundations for channel-based transport costs, including a quadratic cost with a Kraus-operator representation, and establishes symmetry, lower bounds, and a modified triangle inequality with a diagonal term. It also introduces a quantum Wasserstein-1 distance on $n$ qubits, connecting it to the trace distance and exploring tensorization, Lipschitz observables, concentration inequalities, and entropy continuity. The results illuminate how geometry, channel theory, and information-theoretic techniques interact in quantum OT, offering tools for concentration, entropy stability, and potential applications in quantum information processing and computation.
Abstract
These notes are based on the lectures given by the second author at the School on Optimal Transport on Quantum Structures at Erdös 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.