Study of Inequalities in Operator Theory and Quantum Information Theory

1. Shannon熵、von Neumann熵、Umegaki熵和Fujii-Kamei相对算子熵在信息论中占有重要地位。我们将它们扩展到希尔伯特空间中的算子通过使用最近在物理学中很重要的Tsallic熵的思想。研究了算子不等式和迹不等式的一些性质。研究量子计算机的算法，特别是计算复杂性是很有趣的。利用初等数论给出了肖尔因子分解算法的精确估计。研究量子物理中的不确定性关系与量子信息理论之间的关系是一个重要的问题。我们通过推广Wigner-Yanase偏态信息来研究不确定性关系。引入广义Fisher信息作为算子的一种迹，给出了统计学中一个重要的广义Cramer-Rao不等式。

1. Shannon entropy, von Neumann entropy, Umegaki entropy and Fujii-Kamei relative operator entropy are important in information theory. We extended them to operators in Hilbert spaces by using ideas of Tsallic entropy which is recently important in physics. Some properties sund as operator inequalities and trace inequalities were investigated.2. Studies of algorithms of quantum computers, in particular, computational complexity are interesting. We gave precise estimation of Shor's factoring algorithm by using elementary number theory.3. It is an important problem to study the relation between uncertainty relations in quantum physics and quantum information theory. We investigated an uncertainty relation by generalizing Wigner-Yanase skew information.4. Introducing generalized Fisher information which is a trace of operator, we showed a generalized Cramer-Rao inequality which is important in statistics.