Many deep, mysterious connections have been observed between collections of mutually unbiased bases (MUBs) and combinatorial designs called $k$-nets (and in particular, between complete collections of MUBs and finite affine - or equivalently: finite projective - planes). Here we introduce the notion of a $k$-net over an algebra $\mathfrak{A}$ and thus provide a common framework for both objects. In the commutative case, we recover (classical) $k$-nets, while choosing $\mathfrak{A} := M_d(\mathbb C)$ leads to collections of MUBs. A common framework allows one to find shared properties and proofs that "inherently work" for both objects. As a first example, we derive a certain rigidity property which was previously shown to hold for $k$-nets that can be completed to affine planes using a completely different, combinatorial argument. For $k$-nets that cannot be completed and for MUBs, this result is new, and, in particular, it implies that the only vectors unbiased to all but $k \leq \sqrt{d}$ bases of a complete collection of MUBs in $\mathbb C^d$ are the elements of the remaining $k$ bases (up to phase factors). In general, this is false when $k$ is just the next integer after $\sqrt{d}$; we present an example of this in every prime-square dimension, demonstrating that the derived bound is tight. As an application of the rigidity result, we prove that if a large enough collection of MUBs constructed from a certain type of group representation (e.g. a construction relying on discrete Weyl operators or generalized Pauli matrices) can be extended to a complete system, then in fact every basis of the completion must come from the same representation. In turn, we use this to show that certain large systems of MUBs cannot be completed.

中文翻译：

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刚性和相互无偏基和 k-net 的通用框架

在相互无偏基 (MUB) 的集合和称为 $k$-nets 的组合设计之间（特别是在 MUB 的完整集合和有限仿射 - 或等效地：有限射影 - 平面之间）之间，已经观察到许多深刻而神秘的联系。在这里，我们在代数 $\mathfrak{A}$ 上引入了 $k$-net 的概念，从而为这两个对象提供了一个通用框架。在可交换的情况下，我们恢复（经典）$k$-nets，同时选择 $\mathfrak{A} := M_d(\mathbb C)$ 导致 MUB 的集合。一个通用框架允许人们找到对两个对象“固有地有效”的共享属性和证明。作为第一个例子，我们推导出了一个特定的刚性属性，之前已经证明它适用于 $k$-nets，可以使用完全不同的组合参数完成仿射平面。对于无法完成的 $k$-nets 和 MUBs，这个结果是新的，特别是，它意味着唯一的向量对除了 $k \leq \sqrt{d}$ 的完整集合的所有基无偏$\mathbb C^d$ 中的 MUB 是其余 $k$ 基的元素（直至相位因子）。通常，当 $k$ 只是 $\sqrt{d}$ 之后的下一个整数时，这是错误的；我们在每个素数平方维度中展示了一个例子，证明导出的界限是紧的。作为刚性结果的应用，我们证明如果从某种类型的群表示（例如依赖于离散 Weyl 算子或广义泡利矩阵的构造）构造的足够大的 MUB 集合可以扩展到一个完整的系统，那么在事实上，完成的每个基础都必须来自相同的表示。反过来，