A geometrical relation between symmetric operators and mutually unbiased operators

TL;DR

This paper explores the geometric relationship between symmetric operators, including SIC POMs, and mutually unbiased operators, extending the understanding of their structure and interrelation in finite plane geometry.

Contribution

It introduces a geometric framework linking symmetric operators and mutually unbiased operators, generalizing SIC POMs and MUBs, with implications for rank-1 cases.

Findings

Established a geometric relation between symmetric and mutually unbiased operators.

Extended the concepts of SIC POMs and MUBs within this geometric framework.

Discussed implications for rank-1 SIC POMs and MUBs.

Abstract

In this work we study the relation between the set of symmetric operators and the set of mutually unbiased operators from finite plane geometry point of view. Here symmetric operators are generalization of symmetric informationally complete probability-operator measurements (SIC POMs), while mutually unbiased operators are the operator generalization of mutually unbiased bases (MUB). We also discuss the implication of this relation to the particular cases of rank-1 SIC POMs and MUB.