Codes and Expansions (CodEx) Seminar

Steve Flammia (Virginia Tech):
A Constructive Approach to Zauner's Conjecture via the Stark Conjectures

In this talk, I will present a construction of symmetric informationally complete POVMs (SIC-POVMs), a special class of quantum measurements whose existence in all dimensions was conjectured by Zauner in 1999. Equivalently, these are maximal sets of \(d^2\) equiangular lines in \(\mathbb{C}^d\). Our approach introduces an explicit mathematical object, the ghost SIC, built from number-theoretic properties of a special modular function, and we show that it is Galois conjugate to an actual SIC. Assuming two conjectures—Stark’s conjecture from algebraic number theory and a special value identity for a modular function—we prove that our construction produces valid SICs in every dimension. To keep the talk accessible, I will also present a simplified special case of our construction that still implies Zauner’s conjecture. Additionally, I will introduce Zauner.jl, our free open-source software package, which we have used to verify results against known solutions and identify new SICs, including previously undiscovered examples in dimension 100. Time permitting, I will discuss higher-rank generalizations, called r-SICs, which reveal deep connections between SIC existence and abelian field extensions in algebraic number theory. This is joint work with Marcus Appleby and Gene Kopp, arXiv:2501.03970.