量子系统Cd⊗Ckd中无偏的最大纠缠基的构造

Abstract

In this paper, we study the concrete construction of mutually unbiased maximally entangled bases in bipartite systems Cd ⊗Ckd (k ∈ Z+). We first analyze and simplify the sufficient and necessary conditions of two maximally entangled bases to be mutually unbiased in Cd ⊗Ckd, then we use matrix forms to illustrate these conditions in C2 ⊗C2k and C3 ⊗C3k and generalize them in Cd ⊗Ckd. Thus we find that the sufficient and necessary conditions of two maximally entangled bases to be mutually unbiased in Cd ⊗Ckd are translating to the conditions of transit matrices Tkd between two orthonormal bases in Ckd satisfy, that is, Tkd can be divided into k2 submatrices of d ×d satisfying same equations. So the problem of constructing mutually unbiased maximally entangle bases in Cd ⊗Ckd (k ∈ Z+) are changing to the choice of transit matrices in Ckd. According the above equations of transit matrices in Ckd, we first construct two transit matrices in C2, C3 and C4, then we find that using any unitary matrix with nonzero entities in Ck to tensor product the above chosen transit matrices in Cd(d= 2;3;4) from left, we can easily get the transit matrices in Ckd. Hence the choice of transit matrices in Ckd is changing to the choice of unitary matrices with nonzero entities in Ck. Until now, the problem of constructing mutually unbiased maximally entangled bases in Cd ⊗Ckd is really simplified to the choice of unitary matrices in Ck. Since the unitary matrices with nonzero entities in Ck are always exist, we can confirm that mutually unbiased maximally entangled bases in Cd ⊗Ckd(d = 2;3; 4;d ∈ Z+) always exist. Based on the two transit matrices we construct in each space in Ckd(d =2;3; 4;k =1;2;3), we want to describe the concrete construction of two mutually unbiased maximally entangled bases in Cd ⊗Ckd(d= 2;3; 4;k = 1;2;3). As examples, we present a pair of mutually unbiased maximally entangled bases in C2⊗C6 and C3⊗C6, whose dimensions are not prime powers.