Starting from a simple estimation problem, here we propose a general approach
for decoding quantum measurements from the perspective of information
extraction. By virtue of the estimation fidelity only, we provide surprisingly
simple characterizations of rank-1 projective measurements, mutually unbiased
measurements, and symmetric informationally complete measurements. Notably, our
conclusions do not rely on any assumption on the rank, purity, or the number of
measurement outcomes, and we do not need bases to start with. Our work
demonstrates that all these elementary quantum measurements are uniquely
determined by their information-extraction capabilities, which are not even
anticipated before. In addition, we offer a new perspective for understanding
noncommutativity and incompatibility from tomographic performances, which also
leads to a universal criterion for detecting quantum incompatibility.
Furthermore, we show that the estimation fidelity can be used to distinguish
inequivalent mutually unbiased bases and symmetric informationally complete
measurements. In the course of study, we introduce the concept of (weighted
complex projective) $1/2$-designs and show that all $1/2$-designs are tied to
symmetric informationally complete measurements, and vice versa.
