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These notes are based on the lectures given by the second author at theSchool on Optimal Transport on Quantum Structures at Erd\"os Center inSeptember 2022. The focus of the exposition is on two recently introducedapproaches on quantum optimal transport: one based on quantum channels asgeneralized transport plans, the other based on the notion ofHamming-Wasserstein distance of order 1 on multiple-qubit systems. The materialis presented in an elementary manner with a focus on the finite-dimensionalsetting.

Quantum Optimal Transport: Quantum Channels and Qubits