Mutually Unbiased Maximally Entangled Bases In Quantum System C^d（?）C^kd

Mutuall unbiased bases and quantum entanglement are central in Quantum Infor-mation Science. Researches about mutually unbiased bases before are mainly talking about mutually unbiased product bases, mutually unbiased unextendible product bases and so on. Recently, there arises a new conception that combines mutually unbiased bases with entangled states in bipartite systems. Problems that combines mutually unbiased bases with quantum entanglement are not only more and more important, but also to be solved urgently.This article is mainly about maximally entangled bases and the unbiased problem between them. Firstly, the construction of maximally entangled bases in Cd(?)Cκd is studied and a systematicly way to construct maximally entangled bases in Cd(?)Cκd is demonstrated. A maximally entangled basis in Cd(?)Cκd has the following form:Secondly, by studing the mutually unbiased maximally entangled bases in Cd(?)Cκd and analyzing the sufficient and necessary condition between maximally entangled bases that are mutually unbiased, the mutually unbiased problem turns out to be finding the transition matrix T which satisfying the following condition:And I make a conclusion that mutually unbiased maximally entangled bases can be obtained in pairs in space Cd(?)Cκd. Explicit examples are shown in C2(?)C6 and C3(?)C6 separately; Thirdly, this article provide a way to get mutually unbiased maximally entangled bases in Cd(?)C2’d’(d’= κd) from mutually unbiased maximally entangled bases in Cd(?)Cκd ; Lastly, I study about the mutually unbiased problem among multiple maximally entangled bases and find out five mutually unbiased maximally entangled bases in C2(?)C4: where j=0,1,2,3；i=0,1.And three mutually unbiased maximally entangled bases in C2(?)C6: where i=0,1,2;n,,m=0,1.In C~2(?)C~4, there exists no other maximally entangled basis that is mutually unbiased to the ones I find out.