Research Of Joint Density Of Eigenvalues Of The Sum Of Hermitian Random Matrices

The main content of the Horn’s problem is that the spectrum sets of two Hermitian matrices are known then determine the spectral set of the sum of two Hermitian matrices.This problem has been solved by Knutson and Tao Zhexuan by establishing inequalities.Zuber is studied this problem from the perspective of probability density distribution.Based on the Horn’s problem,this paper studies the joint density distribution function of the eigenvalues of the Hermitian random matrix from the perspective of probability density distribution along the Zuber research direction.Firstly,the probability density distribution functions of the diagonal elements of several Hermitian random matrices are given,and then the spectral density of the sum of the Hermitian random matrix is obtained according to the derivative principle.Finally,a special case of the conclusion is applied in the field of quantum information,i.e.the research of the property of the mixed state of quantum states.This thesis includes four chapters.In chapter 1,we give the description of notation used in the paper,and then we explain the research direction and progress of the relevant content in the current context.Finally,we show the introduction of related propositions and theorem conclusions.In chapter 2,we generalizes the conclusion of Zuber to the case of the sum of three Hermitian random matrix.Firstly,the form of probability density function of two Hermi-tian random matrices and diagonal elements is extended to the case of the sum of three Hermitian matrix.Then,the distribution function of the spectral density of the two Her-mitian matrices given by Zuber is extended to the case of the sum of three Hermitian random matrix by using the derivative principle.In chapter 3,we generalizes the joint probability distribution function of diagonal elements and eigenvalues to the case of any finite number of the sum of Hermitian random matrix.In chapter 4,we give an application of this conclusion in the field of quantum infor-mation.It mainly takes the Hermitian random matrix with trace 1 and gives the matrix dimension is 2 and the matrix number is 3.The general expression of the joint probabil-ity density function of the diagonal elements and eigenvalues after the equal probability mixing of two-dimensional quantum states,and the average entropy is studied.