量子系统C^dC^(kd)中无偏的最大纠缠基的构造  

The concrete construction of mutually unbiased maximally entangled bases in C^dC^(kd)

作　　者：陶元红[1,2] 杨强[2] 张军[2] 南华[2] 李林松[2] 

机构地区：[1]首都师范大学数学科学学院,北京100048 [2]延边大学理学院数学系,延吉133002

出　　处：《中国科学：物理学、力学、天文学》2015年第6期74-82,共9页Scientia Sinica Physica,Mechanica & Astronomica

基　　金：国家自然科学基金(批准号:11361065);吉林省自然科学基金(编号201215239)资助项目

摘　　要：本文研究了两体系统CdCkd(k∈Z+)中无偏的最大纠缠基的构造方法.首先利用无偏基的定义分析了CdCkd中两个最大纠缠基无偏的充分必要条件,然后利用此条件将CdCkd中无偏基的构造问题简化成Ck空间中幺正矩阵的选择问题,进而证明了CdCkd(d=2,3,4;k∈Z+)中的无偏的最大纠缠基的存在性,并给出了两个非素数幂维系统C2C6和C3C6中无偏最大纠缠基的具体形式.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 C^d satisfy, that is, Tkd can be divided into k2 submatrices of d x 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;k ∈ 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 pri

关 键 词：无偏基 最大纠缠态 幺正矩阵 

分 类 号：O413.1[理学—理论物理]