Zauner’s Conjecture (existence of SIC-POVMs in all dimensions)

Construct, for every integer d ≥ 1, an equiangular tight frame in C^d consisting of n = d^2 unit vectors (i.e., a SIC-POVM).

Background

Equiangular tight frames with n = d2 in Cd (SIC-POVMs) are central in quantum information and finite frame theory. A recent conditional approach shows that certain number-theoretic conjectures (Stark conjectures) would imply Zauner’s conjecture, but an unconditional proof remains open.

Resolving Zauner’s conjecture would settle decades of work on symmetric informationally complete measurements.

References

It is a well known conjecture that they exist for all dimensions $d$, known as Zauner's Conjecture. For each dimension $d$, there exists an ETF in $\mathbb{C}d$ with $n=d2$ vectors.

— Randomstrasse101: Open Problems of 2025  (2603.29571 - Bandeira et al., 31 Mar 2026) in Conjecture, Section “Mutually Unbiased Bases, ETFs, and Zauner’s Conjecture (ASB)” (Entry 11)