The classical Monge–Kantorovich (MK) problem as originally posed is concerned with how best to move a pile of soil or rubble to an excavation or fill with the least amount of work relative to some cost function. When the cost is given by the square of the Euclidean distance, one can define a metric on densities called the Wasserstein distance. In this note, we formulate a natural matrix counterpart of the MK problem for positive-definite density matrices. We prove a number of results about this metric including showing that it can be formulated as a convex optimisation problem, strong duality, an analogue of the Poincaré–Wirtinger inequality and a Lax–Hopf–Oleinik–type result.

中文翻译：

---

关于矩阵 Monge-Kantorovich 问题

最初提出的经典 Monge-Kantorovich (MK) 问题是关于如何最好地将一堆土壤或碎石移动到挖掘或填充相对于某些成本函数的最少工作量。当成本由欧几里得距离的平方给出时，可以定义一个密度度量，称为瓦瑟斯坦距离. 在这篇笔记中，我们为正定密度矩阵制定了 MK 问题的自然矩阵对应物。我们证明了有关该度量的许多结果，包括表明它可以表述为凸优化问题、强对偶性、Poincaré-Wirtinger 不等式的类似物和 Lax-Hopf-Oleinik 型结果。