Properties And Applications Of F-von Neumann Relative Entropy

Entropy is one of the key concepts of quantum information theory,it is used to measure the physical state of the system contains uncertainty.In quantum infor-mation,the uncertainty of the source of information needs to be characterized by using von Neumann entropy.At the same time,von Neumann entropy also plays an important role in judging quantum entanglement and quantum correlation.Quan-tum relative entropy is a derivative of quantum entropy and plays a central role in information theory.In this paper,we generalize the classical von Neumann relative entropy and von Neumann entropy to the f-von Neumann relative entropy and the f-von Neumann entropy induced by a logarithm-like function f,respectively,and explore their properties.The article is divided into three chapters,the specific structure is as follows:The first chapter introduces the background and research situation of the topic of this paper,and introduces some basic concepts and theorems used later.The second chapter is divided into two sections.In the first section,the defini tion of a logarithm-like function is given,which is the generalization of logarithimic function,a method of constructing logarithn-like function is introduced and some properties of a logarithm-like function are provced.In the second section,the concept,of the f-von Neumnann relative entrolpy is introduced as a generalization of the von Neumnann relative entropy.By defining a special type of logarithm-like funcntion--super logarithm-like function,the non-negative of the f-von Neumann relative entropy is proved.In the third chapter,the concept,of f-von Neumann entropy is given first.By using the non-negative of the f-von Neumann relative entropy,it is then proved that,the f-von Neumann entropy is non-negative,bounded,concave,and subadditive.lastly,stability and continuity of S(p)with respect to the trace distance are proved and some inequalities for the f-von Neumann entropy are also obtained.