Some Research About Uncertainty Relations

Heisenberg uncertainty relation is an important content in quantum mechan-ics and quantum information, and is a hot issue of mathematics and information theory. Based on the quantum theory as background, the integrated use of the think-ing method of analysis, algebra, geometry and topology, and the use of information theory knowledge such as operator algebra, operator theory and matrix analysis, Heisenberg uncertainty relation is systematically researched and generalized by ex-tending the self-adjoint operator to the Hilbert-Schmidt operator. Finally, we study the generalized Wigner-Yanase skew information, discuss its properties, and prove some trace inequalities. The paper is divided into three chapters and the specifics are as follows.In the chapter 1, we mainly introduce some research status and background on our contents, lise some basic concepts, and give the article research direction.In the chapter 2, we discuss the generalized Heisenberg uncertainty relation. Firstly, we introduce some basic concepts such as the symmetrized commutator, the anti-symmetrized commutator, the generalized skew information and the amount of correlation on the Hilbert space; secondly, we give a new generalized Heisen-berg uncertainty relation and its further generalization for a quantity representing a quantum uncertainty.In the chapter 3, we first introduce the definition of the generalized Wigner-Yanase information and discuss its properties; next, we give some trace inequalities related to the uncertainty relation of the generalized Wigner-Yanase information.