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量子系统C<sup><italic>d</italic></sup>⊗C<sup><italic>kd</italic></sup>中无偏的最大纠缠基的构造

Linsong Li、Hua NAN、Qiang Yang、Jun Zhang、YuanHong TAO

2015Zhongguo kexue. Wulixue Lixue TianwenxueMathematics被引 3

摘要

In this paper, we study the concrete construction of mutually unbiased maximally entangled bases in bipartite systems C^{<italic>d</italic>} ⊗C^{<italic>kd</italic>} (<italic>k</italic> ∈ <italic>Z</italic>⁺). We first analyze and simplify the sufficient and necessary conditions of two maximally entangled bases to be mutually unbiased in C^{<italic>d</italic>} ⊗C^{<italic>kd</italic>}, then we use matrix forms to illustrate these conditions in C² ⊗C^2k and C³ ⊗C^3k and generalize them in C^{<italic>d</italic>} ⊗C^{<italic>kd</italic>}. Thus we find that the sufficient and necessary conditions of two maximally entangled bases to be mutually unbiased in C^{<italic>d</italic>} ⊗C^{<italic>kd</italic>} are translating to the conditions of transit matrices <italic>T</italic>_{<italic>kd</italic>} between two orthonormal bases in C^{<italic>kd</italic>} satisfy, that is, <italic>T</italic>_{<italic>kd</italic>} can be divided into <italic>k</italic>² submatrices of <italic>d</italic> ×<italic>d</italic> satisfying same equations. So the problem of constructing mutually unbiased maximally entangle bases in C^{<italic>d</italic>} ⊗C^{<italic>kd</italic>} (<italic>k</italic> ∈ <italic>Z</italic>⁺) 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 C^{<italic>k</italic>} to tensor product the above chosen transit matrices in C^{<italic>d</italic>}(<italic>d</italic>= 2;3;4) from left, we can easily get the transit matrices in C^{<italic>kd</italic>}. Hence the choice of transit matrices in C^{<italic>kd</italic>} is changing to the choice of unitary ma

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DOI：https://doi.org/10.1360/sspma2015-00056