Suppose that for some unit vectors b1,…bn in ℂd we have that for any j≠k bj is either orthogonal to bk or |〈bj,bk〉|2 = 1/d (i.e., bj and bk are unbiased). We prove that if n = d(d + 1), then these vectors necessarily form a complete system of mutually unbiased bases, that is, they can be arranged into d + 1 orthonormal bases, all being mutually unbiased with respect to each other.

中文翻译：

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互无偏基完备系统的一个刚性性质

假设对于一些单位向量b1,…bn在ℂd我们有任何j≠ķ bj要么正交于bķ或者|〈bj,bķ〉|2 = 1/d（IE，bj和bķ是公正的）。我们证明如果n = d(d + 1)，那么这些向量必然形成一个完整的互无偏基系统，即它们可以排列成d + 1正交基，所有基都相互无偏。