On Quantum Optimal Transport

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