A Constructive Approach to Zauner's Conjecture via the Stark Conjectures

• URL: http://arxiv.org/abs/2501.03970v2
• Date: Mon, 17 Mar 2025 12:35:12 GMT
• Title: A Constructive Approach to Zauner's Conjecture via the Stark Conjectures
• Authors: Marcus Appleby, Steven T Flammia, Gene S Kopp,
• Abstract summary: We give a construction of complex equiangular lines in $mathbbCd$, also known as SICPOVMs, conjectured by Zauner to exist for all d.<n>The construction gives a putatively complete list of SICs with Weyl-Heisenberg symmetry in all dimensions d > 3.
• Score: 0.0
• License: http://creativecommons.org/licenses/by-sa/4.0/
• Abstract: We propose a construction of $d^2$ complex equiangular lines in $\mathbb{C}^d$, 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 ${\rm SL}_2(\mathbb{Z})$. 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 $\mathbb{Q}(\sqrt{\Delta})$ where $\Delta = (d+1)(d-3)$, 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 $(d^2-1)$. 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.

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