Unextendible Maximally Entangled Bases And Mutually Bases In Quantum System

The theory of quantum entanglement is an important research topic in the field of quantum information and quantum computing, and it has important applications in many ways.This paper firstly introduces some linear algebra knowledge under the quantum information background, and show mathematics tool relating with the quantum information. For instance, inner product, outside product, tensor product and Pauli matrix, these mathematics knowledge is a tool specially used in research of the quantum information.Then this paper studies the unextendible maximally entangled bases and the problems of construction in low dimension quantum system. Using Pauli matrix, UMEBs are constructed in C2(?)C3 and C2(?)C5, and the method of constructing a pair of unextendible maximally entangled bases is generalized to the Cd(?)Cd’(d’= qd+r, q,r∈Z+,r<d), and show that there are qd2-member UMEBs in Cd(?)Cd’:The paper finally studies Mutually unbiased unextendible maximally entangled bases. We get two groups of unextendible maximally entangled bases by orthonormal basis of C3 in C2(?)C3. The use of unitary matrix{Unm} and three product state constitute a set of complete UMEB. We get another set of basis of C4 through the transition matrix, and show there are two groups of Mutually unbiased unextendible maximally entangled bases. Then two groups of unextendible maximally entangled bases are obtained by a family of unitary matrix in C3(?)C8. Then a general approach to construct mutually unbiased unextendible maximally entangled bases in Cd(?)C2’d’(l∈Z+) from MUUMEBs in Cd(?)Cd’(d’=(d’=qd+r,0<r<d) have been presented.