The Construction Methods Of Unextendible Product Bases

Unextendible Product Bases(U P Bs) occupy an important position in the theory of quantum information, for example, they were introduced as a tool for constructing bound entangled states, they can also be used to construct indecomposible positive maps. Therefore, the problems of UPBs.structure have been widespreadly concerned and a large number of results have been researched. Based on these researches, several methods of the construction of the U P Bs are researched in this paper.The first chapter is the introduction, in which the research status of the unextendible product bases and the main work of this paper are mainly introduced.The basics of U P Bs are mainly introduced in the second chapter, including the Hilbert space, state vector, the representation of state vector of Dirac’s symbolic operations, the relationship of the matrices,and the operations between left and right vectors.Then, some commonly used operators and its operation in Hilbert space are introduced.Finally, the direct product space and its operations are presented.The third chapter is the main content of this paper: the construction of U P Bs in di?erent qubit and two important U P Bs are provided, including Gen Tiles1, GenTiles2. This thesis focuses on the following works: firstly, two di?erent forms of U P Bs in Cm Cnare introduced; secondly,(n- 1)2states UPBs in Cn Cnare constructed;thirdly, another kind of(n-1)2states U P Bs and(n-1)2+1 states U P Bs in Cn Cnare constructed by using graphic analysis; finally, the U P Bs in Cm Cn Csare derived.