Dimension towers of SICs. II. Some constructions

TL;DR

This paper explores a method to construct higher-dimensional SICs from existing ones, focusing on odd dimensions, and provides explicit examples to illustrate the approach, although full proof of SIC properties remains open.

Contribution

It introduces a recipe for constructing aligned SICs in dimension d(d-2) from a given SIC in dimension d, with detailed examples and analysis.

Findings

Constructed sets of vectors in dimension d(d-2) sharing SIC properties

Identified free parameters in the construction process

Provided examples to guide future improvements

Abstract

A SIC is a maximal equiangular tight frame in a finite dimensional Hilbert space. Given a SIC in dimension $d$, there is good evidence that there always exists an aligned SIC in dimension $d(d-2)$, having predictable symmetries and smaller equiangular tight frames embedded in them. We provide a recipe for how to calculate sets of vectors in dimension $d(d-2)$ that share these properties. They consist of maximally entangled vectors in certain subspaces defined by the numbers entering the $d$ dimensional SIC. However, the construction contains free parameters and we have not proven that they can always be chosen so that one of these sets of vectors is a SIC. We give some worked examples that, we hope, may suggest to the reader how our construction can be improved. For simplicity we restrict ourselves to the case of odd dimensions.