In this paper, we describe a possible generalization of the Wasserstein-2 metric, originally defined on the space of scalar probability densities, to the space of Hermitian matrices with trace one and to the space of matrix-valued probability densities. Our approach follows a control-theoretic optimization formulation of the Wasserstein-2 metric, having its roots in fluid dynamics, and utilizes certain results from the quantum mechanics of open systems, in particular the Lindblad equation. It allows determining the gradient flow for the quantum entropy relative to this matricial Wasserstein metric.

中文翻译：

---

矩阵最优传质：量子力学方法


在本文中，我们描述了 Wasserstein-2 度量的可能推广，最初定义在标量概率密度空间上，到具有迹 1 的 Hermitian 矩阵空间和矩阵值概率密度空间。我们的方法遵循 Wasserstein-2 度量的控制理论优化公式，其根源在于流体动力学，并利用开放系统量子力学的某些结果，特别是 Lindblad 方程。它允许确定相对于该矩阵 Wasserstein 度量的量子熵的梯度流。