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 $d\ge4$. 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 $c_-=κ$; 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 $p\equiv5\pmod 6$ or of $3$, an infinite family, we prove that under the algebraic part of Stark's conjecture $c_-=κ$ is equivalent to the triviality of $κ$ 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 $GL_2$ reciprocity law for the squared ghost overlaps that implies $c_-=κ$, 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.

The paper studies the remaining step in the Stark-based approach to Zauner’s conjecture: converting algebraic “ghost SICs,” known to exist for every $d\ge4$, into Hermitian Weyl–Heisenberg-covariant SIC projectors. It formulates this step through Galois actions on a complexified SIC variety and on normalized ghost overlaps. A sufficient condition is a place law: overlap units must be real above one infinite place of a real quadratic field $F$ and unimodular above the other. When the unit field $M/F$ is abelian, this is equivalent to $c_-=\kappa$, identifying complex conjugation at the second place with simultaneous inversion of the units. The paper gives finite-field and single-generating-unit criteria, proves that extended Clifford transformations cannot convert ghosts into SICs, and relates the place law to unitarity of pseudo-unitary Galois conjugates of associated near-group fusion categories. For $d=p^k$ with $p\equiv5\pmod6$, or $d=3^k$ with $k\ge2$, additional algebraic hypotheses reduce the identity to triviality of inversion on $M\cap H_F$. Numerical homotopy continuation finds $512$, $4000$, and $6912$ isolated complexified solutions for $d=4,5,6$, respectively, exactly half Hermitian. Class-field calculations test the place law in $d=4,5,9,13$; an explicit reciprocity law is certified for computed overlaps in $d=5,7,11,15$ using Artin reciprocity and ray-class checks. No new SIC dimension or infinite family is constructed. The implication from Stark’s conjectures was already known; the remaining algebraic Stark and multiplet hypotheses are unproved. Solution counts are numerical, and algebraic certifications depend on numerically identified numbers.