| In this paper,the existence of the mutually unbiased bases(MUBs)and mu-tually unbiased measurements(MUMs)are given in quantum system in the firstly,and we obtain some structure and properties of the MUBs and MUMs,by using the operator algebra and matrix method.Some ways of finding MUBs are given and some structure of MUMs by Bloch sphere are obtained.Secondly,the matrix repre-sentation of bounded linear operators in the two-body quantum system is studied,and we define new matrix representation of bounded linear operators,namely four-dimensional matrix by the orthonormal basis of two-body quantum systems,and studying the relationship of the four-dimensional matrix between bounded linear operators T,S and TS;Finally we study four-dimensional matrix of pure state in the two body-quantum system,and give the relationship of four-dimensional matrix the pure state between the first system and the second system of two-body quantum system.The paper is divided into three chapters and the specifics are as follows:In chapter 1,we mainly introduced some research status and background on our contents,and list some basic concepts and symbols.In chapter 2,Firstly,we give the existence of the mutually unbiased bases(MUBs)in quantum system and obtain some ways of finding MUBs which are illustrated in detail by an example;Secondly,it is obtained the existence of mutually unbiased measurements(MUMs);Finally some structure of MUMs by Bloch sphere are obtained.In chapter 3,In this part,firstly,we give a definition of a new matrix repre-sentation of bounded linear operator T in two-body quantum system,namely four-dimensional matrix;Secondly,the relationship between four-dimensional matrix rep-resentation of bounded linear operators T、S and TS is obtained,and illustrate the results through typical examples;Finally we study the four-dimensional matrix of quantum states,and obtain the relationship between four-dimensional matrix of the first system and the second system of a pure stat,e in two-body system. |
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