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. We construct the desired thermal operation and a heat bath which emulates any Gibbs-preserving map with arbitrary accuracy.

中文翻译：

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关于d-多数化和热-主要化的两个建设性证明

提供了关于d主化和热主化的两个建设性证明。在第一部分中，我们给出了d简化与适当随机矩阵的存在之间的等价关系的图解证明。我们通过使用图形参数显式构造所需的随机矩阵。在第二部分中，我们给出了经典系统中Gibbs保留图和热运算之间的等效性的建设性证明。我们构造所需的热操作和一个热浴，以任意精度模拟任何保存Gibbs的地图。