Two constructive proofs on d-majorization and thermo-majorization

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Two constructive proofs on d-majorization and thermo-majorization are provided.

• In the first part, we present a diagrammatic proof of the equivalence between d-majorization and the existence of a proper stochastic matrix.
• We explicitly construct the desired stochastic matrix by using a graphical argument.
• In the second part, we present a constructive proof of the equivalence between the Gibbs-preserving map and thermal operation in classical systems.