# Tight Frames, Hadamard Matrices and Zauner's Conjecture

## TL;DR

This paper explores the mathematical structures linked to SIC-POVMs, revealing new connections to Hadamard matrices, ETFs, and fusion frames, and investigates their geometric properties and existence conditions.

## Contribution

It introduces novel associations between SICs and various high-dimensional structures, and proposes relaxations and reformulations of the SIC existence problem.

## Key findings

- Structures associated with SICs form continuous manifolds.
- Some manifolds are non-linear, indicating complex geometric structures.
- Relaxations of the SIC existence problem are proposed, reducing equation complexity.

## Abstract

We show that naturally associated to a SIC (symmetric informationally complete positive operator valued measure or SIC-POVM) in dimension d there are a number of higher dimensional structures: specifically a projector and a complex Hadamard matrix in dimension d squared and a pair of ETFs (equiangular tight frames) in dimensions d(d-1)/2, d(d+1)/2. We also show that a WH (Weyl Heisenberg covariant) SIC in odd dimension d is naturally associated to a pair of symmetric tight fusion frames in dimension d. We deduce two relaxations of the WH SIC existence problem. We also find a reformulation of the problem in which the number of equations is fewer than the number of variables. Finally, we show that in at least four cases the structures associated to a SIC lie on continuous manifolds of such structures. In two of these cases the manifolds are non-linear. Restricted defect calculations are consistent with this being true for the structures associated to every known SIC with d between 3 and 16, suggesting it may be true for all d greater than 2.