Study of inequalities and variational principle in free probability theory

在这个研究项目的过程中，我们在自由概率、量子概率和算符/矩阵分析三个学科上取得了不同的结果。我们研究了与自由熵和自由Fisher信息相关的自由概率理论。在量子概率和量子信息学中，我们的研究方向是量子相对熵、量子假设检验和单调黎曼度规。还得到了关于算子单调与凸函数、矩阵迹与范数、算子平均与对称(反)范数等矩阵分析中的几个结果。

In the course of this research project we obtained various results in the three subjects of free probability, quantum probability and operator/matrix analysis. We studied free probability theory related to free entropy and free Fisher information. In quantum probability and quantum information, our study was directed to quantum relative entropy, quantum hypothesis testing and monotone Riemannian metrics. Several results in matrix analysis were also obtained concerning operator monotone and convex functions, matrix trace and norm functions, operator means and symmetric (anti) norms, etc.