Quantum Monge-Kantorovich problem and transport distance between density matrices

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.

arXiv comment15 pages including appendices, 4 figures. Version v2 includes a new quantity, SWAP-fidelity, and some applications

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• Improving the speed of variational quantum algorithms for quantum error correction
• Quantum Wasserstein distance based on an optimization over separable states
• Quantum Earth mover's distance, No-go Quantum Kantorovich-Rubinstein theorem, and Quantum Marginal Problem
• Threshold size for the emergence of a classical-like behaviour
• The Wasserstein distance of order 1 for quantum spin systems on infinite lattices
• On the metric property of quantum Wasserstein divergences
• Ricci curvature of quantum channels on non-commutative transportation metric spaces
• Order $p$ quantum Wasserstein distances from couplings
• Classical shadows meet quantum optimal mass transport
• Isometries of the qubit state space with respect to quantum Wasserstein distances
• Quantum Wasserstein distance between unitary operations
• A perspective on Lindblad's Non-Equilibrium Entropy