Construction Of Several Types Of Mutually Unbiased Bases In Bipartite Quantum System

Mutually unbiased bases and quantum entangled are two important topics in quantum mechanics and quantum information.In the early years,people studied mutually unbiased product bases and unextendible product bases.Later,people studied mutually unbiased maximally entangled bases and unextendible maximally entangled bases.Recently,people began to study mutually unbiased special entangled bases.Since mutually unbiased bases and quantum entangled have a close relation with many problems in quantum information,the study of mutually unbiased entangled bases becomes more and more important.The dissertation is devoted to constructing three types of mutually unbiased bases in bipartite quantum system.The main results are stated as follows.Firstly,we construct special entangled bases determined by Schmidt number 2d in C2d+1(?)C4d.A necessary and sufficient condition for two special entangled bases mutually unbiased are given.Examples of mutually unbiased special entangled bases determined by Schmidt numbers 2 and 4 in C3(?)C4 and C5(?)C8 are provided.Secondly,we construct mutually unbiased unextendible maximally entangled bases in Cd(?)Ckd.We give a necessary and sufficient condition for two sets of complete bases extended from two corresponding unextendible maximally entangled states are mutually unbiased.Examples of mutually unbiased unextendible maximally entangled unbiased bases in C2(?)C4 and C3(?)C6 are provided.Finally,we construct mutually unbiased maximally entangled bases(MUMEB)in Cd(?)Cd.As an example,a set of MUMEBs containing six bases are given in C3(?)C3.