Joint Measurement,Uncertainty Relation Of Random Obsevables And Other Related Studies

In this thesis,we study the joint measurability of 2-outcome unbiased measurement and 3-outcome unbiased measurement,give the rederivation of integral representation of two-qubit Werner state,and discuss the statistical properties of uncertainty relation of random observables.The present thesis is divided into the following four parts:The first chapter is the introduction,which firstly describes the research status of the joint measurability of quantum measurement,the problems related to qubit state,and the uncertainty relation of observables.Secondly,some symbols are explained.Finally,some basic concepts and known propositions needed in this paper are introduced.The second chapter is the joint measurability of 2-outcome unbiased measurement and 3-outcome unbiased measurement.We study the joint measurability of 2-outcome unbiased measurement and 3-outcome unbiased measurement on the qubit system.We obtain a specific condition of their joint measurability by using the sufficient and necessary condition of the joint measurability of POVMs.This sufficient and necessary condition is proposed by Jae et al.We also give a specific example to test the joint measurability condition of 2-outcome unbiased measurement and 3-outcome unbiased measurement.The third chapter is the integral representation of operators.In view of the twoqubit state,we give the rederivation of the integral representation of Werner state.We also calculate the integral representation of three-qubit state.The fourth chapter discusses the uncertainty relation between random observables.Based on the expression of Robertson-Schr ¨odinger uncertainty relation,this chapter mainly studies two problems.To the first question,we discuss the uncertainty-product and its corresponding lower bound on the Gaussian Unitary Ensemble,and we observe the relationship between the gap of their expectations and the dimension;To the second question,we discuss the uncertainty-product and its corresponding lower bound on the Wishart ensemble,and we observe the relationship between the gap of their expectations and the dimension.Finally,we compare conclusions on the Gaussian Unitary Ensemble and the Wishart ensemble.