# The Varieties of Minimal Tomographically Complete Measurements

## TL;DR

This paper investigates the properties, constructions, and mathematical structures of minimal informationally complete quantum measurements (MICs), revealing their complex nature and relation to classical intuitions, and presenting new examples and theoretical insights.

## Contribution

It establishes general properties of MICs, explores their Gram matrices, and provides the first example of an unbiased rank-1 MIC that is not group covariant.

## Key findings

- Characterization of unbiased MICs
- Relation of Gram matrices to linear algebra and number theory
- Discovery of a non-group covariant unbiased rank-1 MIC

## Abstract

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.