2025/01/07 by Marcus Appleby, Appleby, Marcus, Steven T. Flammia +5 · 2 voices · 8 citations
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Coding theory and cryptography #Combinatorics #Computer science #Conjecture #Constructive #Constructive proof #Finite Group Theory Research #Mathematical economics #Mathematics #Programming language #Sociology #math.CO #math.MG #math.NT #quant-ph
paper · pdf · doi:10.48550/arxiv.2501.03970
published in arXiv (Cornell University) (Cornell University)
openalex publication_date 2025/01/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We propose a construction of d2 complex equiangular lines in ℂd, also known as SICPOVMs, conjectured by Zauner to exist for all d. The construction gives a putatively complete list of SICs with Weyl-Heisenberg symmetry in all dimensions d > 3. Specifically, we give an explicit expression for an object we call a ghost SIC, which is constructed from the real multiplication values of a special function and which is Galois conjugate to a SIC. The special function, the Shintani-Faddeev modular cocycle, is more precisely a tuple of meromorphic functions indexed by a congruence subgroup of \rm SL2(ℤ). We prove our construction gives a valid SIC in every case assuming two conjectures: the order 1 abelian Stark conjecture for real quadratic fields and a special value identity for the Shintani-Faddeev modular cocycle. The former allows us to prove that the ghost and the SIC are Galois conjugate over an extension of ℚ(√Δ) where Δ= (d+1)(d-3), while the latter allows us to prove idempotency of the presumptive fiducial projector. We provide computational tests of our SIC construction by cross-validating it with known solutions, particularly the extensive work of Scott and Grassl, and by constructing four numerical examples of nonequivalent SICs in d=100, three of which are new. We further consider rank-r generalizations called r-SICs given by maximal equichordal configurations of r-dimensional complex subspaces. We give similar conditional constructions for r-SICs for all r, d such that r(d-r) divides (d2-1). Finally, we study the structure of the field extensions conjecturally generated by the r-SICs. If K is any real quadratic field, then either every abelian Galois extension of K, or else every abelian extension for which 2 is unramified, is generated by our construction; the former holds for a positive density of field discriminants.