How can d + k vectors in ℝ d be arranged so that they are as close to orthogonal as possible? In particular, define θ ( d , k ) := min X max x ≠ y ∈ X |〈 x , y 〉 | where the minimum is taken over all collections of d + k unit vectors X ⊆ ℝ d . In this paper, we focus on the case here k is fixed and d → ∞. In establishing bounds on θ ( d , k ), we find an intimate connection to the existence of systems of $$\left(\begin{array}{c}k+1\\ 2\end{array}\right)$$ ( k + 1 2 ) equiangular lines in ℝ k . Using this connection, we are able to pin down θ ( d , k ) whenever k ∈ {1, 2, 3, 7, 23} and establish asymptotics for general k . The main tool is an upper bound on $$\mathbb{E}_{x,y\sim\mu}|\langle{x,y}\rangle|$$ E x , y ~ µ | ⟨ x , y ⟩ | whenever μ is an isotropic probability mass on ℝ k , which may be of independent interest. Our results translate naturally to the analogous question in ℂ d . In this case, the question relates to the existence of systems of k 2 equiangular lines in ℂ k , also known as SIC-POVM in physics literature.

中文翻译：

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近正交向量和小对映球码

ℝ d 中的 d + k 个向量如何排列，使它们尽可能接近正交？特别地，定义 θ ( d , k ) := min X max x ≠ y ∈ X |〈 x , y 〉 | 其中最小值取自 d + k 个单位向量 X ⊆ ℝ d 的所有集合。在本文中，我们关注 k 固定且 d → ∞ 的情况。在建立 θ ( d , k ) 的边界时，我们发现 $$\left(\begin{array}{c}k+1\\ 2\end{array}\right)$ 系统存在的密切联系$ ( k + 1 2 ) ℝ k 中的等角线。使用这种联系，我们能够在 k ∈ {1, 2, 3, 7, 23} 时确定 θ ( d , k ) 并建立一般 k 的渐近性。主要工具是 $$\mathbb{E}_{x,y\sim\mu}|\langle{x,y}\rangle|$$ E x , y ~ µ | 的上限 ⟨ x , y ⟩ | 每当 μ 是 ℝ k 上的各向同性概率质量时，这可能是独立感兴趣的。我们的结果很自然地转化为 ℂ d 中的类似问题。在这种情况下，问题与 ℂ k 中 k 2 等角线系统的存在有关，在物理学文献中也称为 SIC-POVM。