Quantum Measurements in the Light of Quantum State Estimation

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.

19+15 pages, 8 figures, and 1 table; published in PRX Quantum

References in corpus (6)

• Tight informationally complete quantum measurements
• Weighted complex projective 2-designs from bases: optimal state determination by orthogonal measurements
• Coexistence of qubit effects
• Quantum state estimation with informationally overcomplete measurements
• Universally Fisher-Symmetric Informationally Complete Measurements
• Tomographic and Lie algebraic significance of generalized symmetric informationally complete measurements

Cited by in corpus (4)

• Experimental Realization of Genuine Three-copy Collective Measurements for Optimal Information Extraction
• Entanglement dynamics and ergodicity breaking in a quantum cellular automaton
• Spin minimum uncertainty states for refined uncertainty relations
• Optimal estimation of three parallel spins with genuine and restricted collective measurements