Tomographic and Lie algebraic significance of generalized symmetric informationally complete measurements

TL;DR

This paper explores the properties of generalized symmetric informationally complete measurements, revealing their unique role in quantum state tomography and Lie algebra, and establishing their optimality and algebraic characterization.

Contribution

It demonstrates that generalized SICs are uniquely characterized by antisymmetry in Lie algebra structure constants and are optimal for quantum state tomography among minimal IC measurements.

Findings

Generalized SICs are optimal in quantum state tomography with fixed average purity.

They are uniquely characterized by antisymmetry in Lie algebra structure constants.

These properties distinguish generalized SICs from other minimal IC measurements.

Abstract

Generalized symmetric informationally complete (SIC) measurements are SIC measurements that are not necessarily rank one. They are interesting originally because of their connection with rank-one SICs. Here we reveal several merits of generalized SICs in connection with quantum state tomography and Lie algebra that are interesting in their own right. These properties uniquely characterize generalized SICs among minimal IC measurements although, on the face of it, they bear little resemblance to the original definition. In particular, we show that in quantum state tomography generalized SICs are optimal among minimal IC measurements with given average purity of measurement outcomes. Besides its significance to the current study, this result may help understand tomographic efficiencies of minimal IC measurements under the influence of noise. When minimal IC measurements are taken as bases for the Lie algebra of the unitary group, generalized SICs are uniquely characterized by the antisymmetry of the associated structure constants.