Some non-standard ways to generate SIC-POVMs in dimensions 2 and 3

TL;DR

This paper introduces new methods for constructing SIC-POVMs in dimensions 2 and 3, showing these are equivalent to known solutions but offering alternative approaches to their generation.

Contribution

It presents novel construction techniques for SIC-POVMs in low dimensions, expanding the toolkit for their generation and analysis.

Findings

Constructed SIC-POVMs are unitarily equivalent to known solutions.

New methods provide alternative ways to generate SIC-POVMs.

Results contribute to understanding the structure of SIC-POVMs in low dimensions.

Abstract

The notion of Symmetric Informationally Complete Positive Operator-Valued Measures (SIC-POVMs) arose in physics as a kind of optimal measurement basis for quantum systems. However the question of their existence is equivalent to that of the existence of a maximal set of \emph{complex equiangular lines}. That is to say, given a complex Hilbert space of dimension $d$, what is the maximal number of (complex) lines one can find which all make a common (real) angle with one another, in the sense that the inner products between unit vectors spanning those lines all have a common absolute value? A maximal set would consist of $d^2$ lines all with a common angle of $\arccos{\frac{1}{\sqrt{d+1}}}$. The same question has been posed in the real case and some partial answers are known. But at the time of writing no unifying theoretical result has been found in the real or the complex case: some sporadic low-dimensional numerical constructions have been converted into algebraic solutions but beyond this very little is known. It is conjectured that such structures always arise as orbits of certain fiducial vectors under the action of the Weyl (or generalised Pauli) group. In this paper we point out some new construction methods in the lowest dimensions ($d=2$ and $d=3$). We should mention that the SIC-POVMs so constructed are all unitarily equivalent to previously known SIC-POVMs.