Analogous to the notion of mutually unbiased bases for Hilbert spaces, we consider mutually unbiased unitary bases (MUUBs) for the space of operators, [Formula: see text], acting on such Hilbert spaces. The notion of MUUB reflects the equiprobable guesses of unitary operators in one basis of [Formula: see text] when estimating a unitary operator in another. Though, for prime dimension [Formula: see text], the maximal number of MUUBs is known to be [Formula: see text], there is no known recipe for constructing them, assuming they exist. However, one can always construct a minimum of three MUUBs, and the maximal number is approached for very large values of [Formula: see text]. MUUBs can also exist for some [Formula: see text]-dimensional subspace of [Formula: see text] with the maximal number being [Formula: see text].

中文翻译：

---

d维希尔伯特空间上算子的互无偏酉基

类似于 Hilbert 空间的互无偏基的概念，我们考虑作用于此类 Hilbert 空间的运算符空间的互无偏酉基 (MUUB) [公式：见正文]。MUUB 的概念反映了酉算子在 [公式：见正文] 的一个基础上估计另一个酉算子时的等概率猜测。虽然，对于素数维度 [公式：见文本]，MUUB 的最大数量已知为 [公式：见文本]，但假设它们存在，则没有已知的构造它们的方法。然而，一个人总是可以构造至少三个 MUUB，并且对于非常大的 [公式：参见文本] 值接近最大数量。MUUBs 也可以存在于 [Formula: see text] 的某些 [Formula: see text] 维子空间，其中最大数量为 [Formula: see text]。