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≥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-=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≡5pmod 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 GL2 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.
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BibTeX
@misc{khodaiemehr2026from,
title = {{From Ghost SICs to SIC-POVMs: A Galois Reduction of Zauner's Conjecture}},
author = {Hassan Khodaiemehr},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01524},
eprint = {2611.01524},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01524},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - From Ghost SICs to SIC-POVMs: A Galois Reduction of Zauner's Conjecture
AU - Hassan Khodaiemehr
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01524
UR - https://arxiv.org/abs/2611.01524
AB - 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≥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_-=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≡5pmod 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.
ER -