Quantum Monge-Kantorovich problem and transport distance between density matrices

TL;DR

This paper introduces a quantum optimal transport framework for density matrices, defining a semidistance related to the Bures distance, with applications in quantum machine learning and a new measure called SWAP-fidelity.

Contribution

It develops a quantum Monge-Kantorovich problem, deriving a quantum transport distance that extends classical Wasserstein metrics to density matrices.

Findings

Minimal transport cost forms a semidistance between quantum states.

Square root of the cost satisfies the triangle inequality.

Introduces SWAP-fidelity for quantum state proximity measurement.

Abstract

A quantum version of the Monge--Kantorovich optimal transport problem is analyzed. The transport cost is minimized over the set of all bipartite coupling states $\rho^{AB}$, such that both of its reduced density matrices $\rho^A$ and $\rho^B$ of dimension $N$ are fixed. We show that, selecting the quantum cost matrix to be proportional to the projector on the antisymmetric subspace, the minimal transport cost leads to a semidistance between $\rho^A$ and $\rho^B$, which is bounded from below by the rescaled Bures distance and from above by the root infidelity. In the single qubit case we provide a semi-analytic expression for the optimal transport cost between any two states and prove that its square root satisfies the triangle inequality and yields an analogue of the Wasserstein distance of order two on the set of density matrices. We introduce an associated measure of proximity of quantum states, called SWAP-fidelity, and discuss its properties and applications in quantum machine learning.