Quantum Measurements in the Light of Quantum State Estimation

TL;DR

This paper introduces a novel approach to quantum measurement characterization based on information extraction, providing simple criteria for key measurement types and insights into quantum incompatibility without relying on traditional assumptions.

Contribution

It offers a new perspective on quantum measurement decoding through estimation fidelity, characterizes elementary measurements without prior assumptions, and links measurement types to complex projective designs.

Findings

Characterization of rank-1 projective, mutually unbiased, and SIC measurements via estimation fidelity.

Universal criterion for quantum incompatibility based on tomographic performance.

Connection between 1/2-designs and symmetric informationally complete measurements.

Abstract

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.