Using character of mutually orthogonal Latin squares, we first prove that two special squares are orthogonal to a complete set of mutually orthogonal Latin squares of order d . Then using the complete set of mutually orthogonal Latin squares of order d and two special squares, we construct d + 1 mutually unbiased bases in ℂ d ⊗ ℂ d $\mathbb {C}^{d} \otimes \mathbb {C}^{d}$ , which include d − 1 mutually unbiased maximally entangled bases and two mutually unbiased product bases. We also present the corresponding construction in ℂ 3 ⊗ ℂ 3 $\mathbb {C}^{3} \otimes \mathbb {C}^{3}$ , ℂ 4 ⊗ ℂ 4 $\mathbb {C}^{4} \otimes \mathbb {C}^{4}$ and ℂ 5 ⊗ ℂ 5 $\mathbb {C}^{5} \otimes \mathbb {C}^{5}$ .

中文翻译：

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使用相互正交拉丁方构建相互无偏基

利用相互正交的拉丁方阵的性质，首先证明两个特殊的方阵正交于一组完整的d阶相互正交的拉丁方阵。然后使用完整的 d 阶相互正交拉丁方阵和两个特殊方阵，我们在 ℂ d ⊗ ℂ d $\mathbb {C}^{d} \otimes \mathbb {C}^ 中构造 d + 1 个互无偏基{d}$ ，其中包括 d − 1 个相互无偏的最大纠缠基和两个相互无偏的乘积基。我们还在 ℂ 3 ⊗ ℂ 3 $\mathbb {C}^{3} \otimes \mathbb {C}^{3}$ , ℂ 4 ⊗ ℂ 4 $\mathbb {C}^{4} 中给出了相应的构造\otimes \mathbb {C}^{4}$ 和 ℂ 5 ⊗ ℂ 5 $\mathbb {C}^{5} \otimes \mathbb {C}^{5}$ 。