Monotone and convex operator functions with applications in quantum information and mathematical physics

证明了各种算子或迹泛函的凸性。这一系列的研究使用了一些方法，包括我们开发的一些算子不等式。我们发现并证明了量子系统的残差熵和一个或多个量子通道上的熵增益的凸性。这一发现是基于算子单调函数理论的最新进展，并发表在《统计物理杂志》上的一篇论文和2013年6月25日至26日在法国巴黎举行的《量子力学中的熵：最新进展》会议上。我们还表征了矩阵熵的类，这是一种最近建立随机矩阵集中不等的工具。我们在单调和凸算子映射理论及其在量子物理中的应用方面获得了新的结果。我们还刻画了非对易观点，并开始研究所谓的正则映射及其在矩阵几何中的应用。

We proved convexity of various operator or trace functionals. This line of research uses methods, including a number of operator inequalities, which we developed. We discovered and proved a convexity property for the residual entropy of a quantum system, and for the entropy gain over one or multiple quantum channels. The discovery is based on recent advances in the theory of operator monotone functions and is reported in a paper published in the Journal of Statistical Physics and at the conference "Entropy in Quantum Mechanics: Recent advances", June 25-26 2013, Paris, France.We also characterised the class of matrix entropies, which is a recent tool to establish concentration inequalities for random matrices.We obtained new results in the theory of monotone and convex operator mappings, with applications in quantum physics. We also characterised the non-commutative perspectives and initiated a study of so-called regular mappings with applications in matrix geometry.