Matrix Exponential,Matrix Logrithm,and Application In Quantum Information Theory

In this paper,we mainly study the finite matrix polynomial for the matrix exponential and the matrix logarithm,and illustrate the applications of the matrix exponential in time evolution of quantum states and the matrix logarithm in the preservation of relative entropy under a quantum channel.This paper is mainly separated into the following four chapters:The first chapter is the introduction,which firstly describes the current research s-tatus of the matrix exponential,the matrix logarithm and its application in quantum in-formation theory.Then,some symbols used in this paper are fixed.Finally,some basic concepts and known propositions are given.The second chapter is the matrix exponential,which mainly gives the matrix expo-nential formula for n×n complex matrix with n distinct eigenvalues.We firstly describe 2×2 complex matrix,we give a new proof of the matrix exponential formula that ob-tained by Foulis recently.Then,we rederive the matrix exponential function formulae of the complex normal matrix and Pauli matrix in the range of[0,2π].Finally,we get the matrix exponential formula of the n×n complex matrix with n distinct eigenvalues.Furthermore,we derive the corresponding formula of general matrix function.The third chapter is the matrix logarithm,which mainly gives the matrix logarithm formula for n×n positive definite matrix with n distinct eigenvalues.Firstly,we give the 2×2 matrix logarithm formula of positive definite matrix with two different eigenvalues,and by using the Bloch representation of a qubit state,we obtain the qubit state’s matrix logarithm formula described by Bloch vector and its length.Then we present a new representation of the Von Neumann entropy and quantum relative entropy of the two qubit states,and further derive that the expectation of the relative entropy of two random qubit states is equal to 1.Finally,the matrix logarithm formula of n×n positive definite matrix with n distinct eigenvalues is obtained.The fourth chapter discusses the application of the matrix exponential and the matrix logarithm in quantum information theory.First,we illustrate the application of the matrix exponential in time evolution of quantum states,for a given Hamiltonian H,the evolution of quantum state is described by e-itHpeitH according to Quantum Mechanics.In partic-ular,for a qubit system,by using the 2×2 matrix exponential formula we can obtain the simple and intuitive representation of the evolved state so as to solve the extremum problem of fidelity between a fixed qubit state and an evolved state.Then,we describe an application of the matrix logarithm in studying the preservation of relative entropy under a qubit channel.Finally,we discuss the relations of the quantum states which can preserve relative entropy under three types of specific qubit channels,i.e.,bit flip channel,phase flip channel and bit-phase flip channel,respectively.