On Quantum Optimal Transport

Abstract

We analyze a quantum version of the Monge–Kantorovich optimal transport problem. The quantum transport cost related to a Hermitian cost matrix C is minimized over the set of all bipartite coupling states ρAB\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho ^{AB}$$\end{document} with fixed reduced density matrices ρA\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho ^A$$\end{document} and ρB\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho ^B$$\end{document} of size m and n. The minimum quantum optimal transport cost TCQ(ρA,ρB)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{T}^Q_{C}(\rho ^A,\rho ^B)$$\end{document} can be efficiently computed using semidefinite programming. In the case m=n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m=n$$\end{document} the cost TCQ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{T}^Q_{C}$$\end{document} gives a semidistance if and only if C is positive semidefinite and vanishes exactly on the subspace of symmetric matrices. Furthermore, if C satisfies the above conditions, then TCQ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sqrt{\textrm{T}^Q_{C}}$$\end{document} induces a quantum analogue of the Wasserstein-2 distance. Taking the quantum cost matrix CQ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^Q$$\end{document} to be the projector on the antisymmetric subspace, we provide a semi-analytic expression for TCQQ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{T}^Q_{C^Q}$$\end{document} for any pair of single-qubit states and show that its square root yields a transport distance on the Bloch ball. Numerical simulations suggest that this property holds also in higher dimensions. Assuming that the cost matrix suffers decoherence and that the density matrices become diagonal, we study the quantum-to-classical transition of the Monge–Kantorovich distance, propose a continuous family of interpolating distances, and demonstrate that the quantum transport is cheaper than the classical one. Furthermore, we introduce a related quantity—the SWAP-fidelity—and compare its properties with the standard Uhlmann–Jozsa fidelity. We also discuss the quantum optimal transport for general d-partite systems.