What are the minimal conditions required to define a SIC POVM?

TL;DR

This paper investigates the minimal conditions for defining SIC POVMs, introduces semi-SIC POVMs, and explores their existence and properties across different dimensions, highlighting open questions in higher-dimensional cases.

Contribution

It clarifies the necessity of the trace condition in SIC POVMs and introduces semi-SIC POVMs, expanding the framework and understanding of quantum measurements.

Findings

Semi-SIC POVMs exist in dimension two and are fully characterized.

The trace condition is essential for SIC POVMs, leading to the definition of semi-SIC POVMs.

Existence of semi-SIC POVMs in higher dimensions remains an open problem.

Abstract

Symmetric informationally complete (SIC) POVMs are a class of quantum measurements which, in addition to being informationally complete, satisfy three conditions: 1) every POVM element is rank one, 2) the Hilbert-Schmidt inner product between any two distinct elements is constant, and 3) the trace of each element is constant. The third condition is often overlooked, since it may give the impression that it follows trivially from the second. We show that this condition cannot be removed, as it leads to two distinct values for the trace of an element of the POVM. This observation has led us to define a broader class of measurements which we call semi-SIC POVMs. In dimension two we show that semi-SIC POVMs exist, and we construct the entire family. In higher dimensions, we characterize key properties and applications of semi-SIC POVMs, and note that the proof of their existence remains open.