Constructions Of Mutually Unbiased Uniform Bases

As the most important physical resource in quantum communication,quantum entanglement is widely utilized in quantum information processing,such as quantum state tomography,quantum error correction code,superdense coding and quantum secret sharing.A k-uniform state of N parties is the maximally mixed for all reductions of k parties and it is a very important class of entangled pure state.Mutually Unbiased Bases is an important concept of quantum information theory.It is of great theoretical significance to investigate its construction.In this paper,we study the construction of mutually unbiased bases in bipartite and multipartite systems based on the 1-uniform state.The following results are obtained:(1)In the bipartite system Cd(?)Cd’,we put forward a method by utilizing the unitary matrix and construct a pair of mutually unbiased 1-uniform bases;(2)We first present a pair of mutually unbiased 1-uniform bases in(Cd)(?)3,then derive a pair of mutually unbiased 1-uniform bases in multipartite homogeneous system through the recursive structure;(3)For general tripartite system (?)i=13Cdi(d1≤d2≤d3),we propose a new construction and obtain a pair of mutually unbiased 1-uniform bases.Moreover,we extend the above construction method to general multipartite system (?)i=1nCdi(d1≤d2 ≤…≤dn)and construct a pair of mutually unbiased 1-uniform bases.