Quantum optimal transport is cheaper

TL;DR

This paper compares classical and quantum bipartite matching problems, demonstrating that quantum optimal transport can be more cost-effective than classical methods, especially with particles of different masses.

Contribution

It introduces a quantum Wasserstein distance framework and shows that quantum costs can be strictly lower than classical costs in certain cases.

Findings

Quantum costs can be cheaper than classical costs.

Equal mass particles have equal quantum and classical costs.

Different mass particles can have strictly lower quantum costs.

Abstract

We compare bipartite (Euclidean) matching problems in classical and quantum mechanics. The quantum case is treated in terms of a quantum version of the Wasserstein distance introduced in [F. Golse, C. Mouhot, T. Paul, Commun. Math. Phys. 343 (2016), 165-205]. We show that the optimal quantum cost can be cheaper than the classical one. We treat in detail the case of two particles: the equal mass case leads to equal quantum and classical costs. Moreover, we show examples with different masses for which the quantum cost is strictly cheaper than the classical cost.