From Ghost SICs to SIC-POVMs: A Galois Reduction of Zauner's Conjecture

Abstract

Symmetric informationally complete positive operator-valued measures (SIC-POVMs, or SICs) are optimal minimal measurements for quantum state tomography. Problem~1 of Horodecki, Rudnicki and Życzkowski's open problems in quantum information asks for SICs in infinitely many dimensions. By work of Appleby--Flammia--Kopp and Radchenko--Wheeler, ghost SICs, which satisfy the SIC equations with Hermiticity up to parity, exist unconditionally for $d\ge 4$. We study the passage from ghosts to SICs. We compute all isolated points of the complexified SIC variety for $d=4,5,6$; exactly half are SICs. We prove that no symmetry of the SIC equations maps a ghost to a SIC, so the step is purely arithmetic, and that a SIC exists in dimension $d$ whenever the ghost overlap units satisfy a place law: each Galois conjugate is real or unimodular according to the infinite place of the real quadratic field $F$ below it. If the unit field is abelian over $F$, the place law is the Stark relation $c_{-}=\kappa$; equivalently, a Galois conjugate of an associated near-group fusion category is unitary exactly when it is pseudo-unitary. For $d$ a power of a prime $p\equiv 5(\mathrm{mod}6)$ or of $3$, an infinite family, we prove that under the algebraic part of Stark's conjecture $c_{-}=\kappa$ is equivalent to the triviality of $\kappa$ on the Hilbert-class-field part of the unit field. That condition involves no complex conjugation, and it follows from a sign-free multiplet hypothesis. We state an explicit $G{L}_2$ reciprocity law for the squared ghost overlaps that implies $c_{-}=\kappa$, and we prove it for the computed algebraic overlaps in $d=5,7,11,15$ by Artin reciprocity and a check in every ray class. Zauner's conjecture remains open: the algebraic part of Stark's conjecture and the multiplet hypothesis are not proved here.

arXiv comment100 pages, 1 figure, 18 tables. Code and data: https://github.com/hassankhodaiemehr/ghost-sic-galois