Minimal Informationally Complete quantum measurements, or MICs, illuminate the structure of quantum theory and how it departs from the classical. Central to this capacity is their role as tomographically complete measurements with the fewest possible number of outcomes for a given finite dimension. Despite their advantages, little is known about them. We establish general properties of MICs, explore constructions of several classes of them, and make some developments to the theory of MIC Gram matrices. These Gram matrices turn out to be a rich subject of inquiry, relating linear algebra, number theory and probability. Among our results are some equivalent conditions for unbiased MICs, a characterization of rank-1 MICs through the Hadamard product, several ways in which immediate properties of MICs capture the abandonment of classical phase space intuitions, and a numerical study of MIC Gram matrix spectra. We also present, to our knowledge, the first example of an unbiased rank-1 MIC which is not group covariant. This work provides further context to the discovery that the symmetric informationally complete quantum measurements (SICs) are in many ways optimal among MICs. In a deep sense, the ideal measurements of quantum physics are not orthogonal bases.

中文翻译：

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最小断层完整测量的种类

最小信息完整的量子测量，或 MIC，阐明了量子理论的结构以及它如何偏离经典。这种能力的核心是它们作为断层完整测量的作用，对于给定的有限维度，结果数量可能最少。尽管它们具有优势，但人们对它们知之甚少。我们建立了MIC的一般性质，探索了MIC的几类构造，并对MIC Gram矩阵的理论进行了一些发展。这些 Gram 矩阵被证明是一个丰富的研究主题，涉及线性代数、数论和概率。我们的结果中有一些无偏 MIC 的等效条件，通过 Hadamard 产品对 1 级 MIC 的表征，MIC 的直接属性捕捉放弃经典相空间直觉的几种方式，以及 MIC Gram 矩阵光谱的数值研究。据我们所知，我们还介绍了第一个无偏 rank-1 MIC 的例子，它不是组协变的。这项工作为发现对称信息完整量子测量 (SIC) 在许多方面在 MIC 中是最佳的这一发现提供了进一步的背景。从更深的意义上说，量子物理学的理想测量不是正交基。