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

点击翻译按钮获取中文摘要

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.