Operator Inequalities Related To Quantum Information

In this dissertation,we study the operator functions,operator inequalities related to quantum physics and quantum information.we discuss the properties of the generalized perspective functions of multivariate regular functions,Lieb-Ruskai's convexity theorem,a new kinds of operator convex functions and their Frechet differentials,Peierls-Bogolyubov inequality as well as operator mean inequalities.The main contents are summarized as follows:In chapter 1,we outline the research background associated with this dissertation including the basic notations,fundamental operator theories and the history related to quantum entropy inequalities and matrix convexity and concavity theorems.Besides,we state the main results and innovation points of this thesis.In chapter 2,we discuss the properties of perspective maps and introduce the generalized perspective functions of multivariate regular functions.We point out some defects in Ebadian-Nikoufar-Gordji's work on the perspective mappings for operator concave(resp.convex)functions and present our correction for them.Furthermore,based on Hansen's conceptions of regular operator mappings and the perspective of a regular mapping,we introduce the generalized perspective associated to a regular operator mapping of several variables and a regular mapping of one variable and discuss its properties.At the end,we use the perspective method in quantum entropy inequalities.In chapter 3,we are concerned with the Lieb-Ruskai's convexity theorem,we give an generalization of the theorem concerned with operator decreasing positive function.We also consider the generalization of Lieb-Ruskai's convexity theorem to multivariables and the equivalent theorems of its monotonicity inequalities.In chapter 4,we work out a new class of operator concave(resp.convex)functions of two variables as well as derive concavity(resp.convexity)of the trace functions associated with the Frechet differential mapping of certain power functions.Furthermore,by using perspective method,operator concave(resp.convex)functions of three or four commuting variables are also investigated.as an application of our theoretical results in the second section,we explore the concavity(resp.convexity)of the Frechet differential mapping x? dg(x)*df(x)-1 in positive definite invertible operators with f(t)=tp for 0<p?1 and g(t)=tq for p?q?p+1.In chapter 5,we establish some new trace inequalities,and generalized the famous Peierls-Bogolyubov inequality.Moreover we establish a variational representation for trace related to the relative entropy and parameterized a concavity theorem of Lieb.In chapter 6,we study operator mean inequalities.Firstly,by using the perspective map of multivariate regular functions we establish the conception of weighted multivariate operator geometric means,and show that the weighted multivariate operator geometric mean possess several attractive properties.Moreover,by using the the Taylor series of some hyperbolic functions we refine the ordering relations among the Heinz means with different parameters and obtain some improvements of the Heinz operator inequalities.At last,we show that the differences of the Mercer's power means Qr,?(a,b,x)s associated to distinct sequences of weights are comparable,in terms of constants that depend on the smallest and largest quotients of the weights.