Construction of general symmetric-informationally-complete-positive-operator-valued measures by using a complete orthogonal basis

TL;DR

This paper introduces a method to construct general symmetric-informationally-complete POVMs using complete orthogonal bases, enhancing quantum state tomography with explicit constructions and spectral analysis.

Contribution

It provides a new systematic construction of GSIC POVMs via complete orthogonal bases, including necessary and sufficient conditions for SIC POVMs in any dimension.

Findings

Spectral properties of COBs are crucial for SIC POVM construction.

Constructive methods for generating COBs from simple bases.

Bounds on mean-square error in quantum state tomography are derived.

Abstract

A general symmetric-informationally-complete (GSIC)-positive-operator-valued measure (POVM) is known to provide an optimal quantum state tomography among minimal IC POVMs with a fixed average purity. In this paper we provide a general construction of a GSIC POVM by means of a complete orthogonal basis (COB), also interpreted as a normal quasiprobability representation. A spectral property of a COB is shown to play a key role in the construction of SIC POVMs and also for the bound of the mean-square error of the state tomography. In particular, a necessary and sufficient condition to construct a SIC POVM for any d is constructively given by the power of traces of a COB. We give three simple constructions of COBs from which one can systematically obtain GSIC POVMs.