Tight frames, Hadamard matrices and Zauner’s conjecture

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, a complex Hadamard matrix in dimension d 2 and a pair of ETFs (equiangular tight frames) in dimensions 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 ≥ 3.

SICs: Some Explanations

“…It is also tempting to believe that one can prove SIC existence for all the dimensions in such a sequence in some inductive way. At the moment this is just a dream, but bits and pieces of what looks like an argument to this effect have materialized [34][35][36][37]. We can simplify the story quite a bit by starting it from an observation by Renes et al [5], and this is what we propose to do here.…”

The numerics of phase retrieval

“…We also know analytic solutions for a few values of n; see e.g. Appleby, Bengtsson, Flammia and Goyeneche (2019) and Fuchs et al (2017).…”

Entropic uncertainty relations from equiangular tight frames and their applications

“…Studies of discrete structures in finite-dimensional Hilbert spaces has a long history [1,2]. Such structures are interesting not only in their own rights but also due to potential application in quantum physics.…”