[#P46] Zauner's conjecture
Problem. For every integer \(d\ge2\), there exist \(d^2\) unit vectors in \(\mathbb{C}^d\) whose pairwise squared inner-product magnitudes are all \(1/(d+1)\).
1Context
The conjecture links optimal quantum measurements with equiangular lines, number fields, and special values of analytic functions.
2Problem setup
Definition 1 (The corresponding rank-one operators form a symmetric informationally complete positive operator-valued measure, or SIC-POVM). The corresponding rank-one operators form a symmetric informationally complete positive operator-valued measure, or SIC-POVM.
Definition 2 (Weyl-Heisenberg covariance). Weyl-Heisenberg covariance means that all d^2 lines arise from one fiducial vector under the discrete displacement operators.
Remark 1. The conjecture links optimal quantum measurements with equiangular lines, number fields, and special values of analytic functions.
3What counts as a solution
- Construct a Weyl-Heisenberg covariant SIC in every dimension d at least 2, or give a dimension and prove that no such SIC exists.
1Status
Current status (Dated status and exact unresolved remainder). Unresolved in this packet after the dated source check. Strongest checked result: The 2025 construction works in every d>3 assuming the order-1 abelian Stark conjecture and a Shintani-Faddeev special-value identity. Finite-dimensional constructions do not prove all dimensions. Exact unresolved remainder: Construct a Weyl-Heisenberg covariant SIC for every d>=2, or prove a dimension has none.[1][2]
1Records
Notes and companion material
Original intake status. The cited 2025 paper treats Weyl-Heisenberg SIC existence in every dimension as Zauner's conjecture and proves a conditional construction. The source and public status were checked on 2026-07-31. This is an admin-curated seed record, not an independent exhaustive literature review.
- Exact and numerical SICs are known in many dimensions. Conditional constructions using Stark-type conjectures do not settle unconditional existence.
Computational notes
- High-precision numerical fiducials provide evidence in particular dimensions but require exact certification and do not cover all dimensions.
2See also
- Lehmer's Mahler measure problemalgebraic number theory
- Artin's primitive root conjecturealgebraic number theory
How to cite
TheoremDB contributors, “Zauner's conjecture,” TheoremDB research memory, snapshot of July 31, 2026. https://theoremdb.org/statements/zauners-conjectureThis page as plain text: zauners-conjecture.md
This problem includes 2 records joined by 2 typed links, sourced from arxiv.org[1], current as of July 31, 2026.
1References
- Packet source. Marcus Appleby, Steven T Flammia, and Gene S Kopp, “A Constructive Approach to Zauner's Conjecture via the Stark Conjectures”. arXiv:2501.03970 (2025). Marcus Appleby, Steven T. Flammia, and Gene S. Kopp, arXiv:2501.03970, abstract and main conjecture. ↗preprint · primary source · arXiv:2501.03970, checked 2026-07-31 · checked 2026-07-31Source use: original summary.The cited 2025 paper treats Weyl-Heisenberg SIC existence in every dimension as Zauner's conjecture and proves a conditional construction. The source and public status were checked on 2026-07-22. This is an admin-curated seed record, not an independent exhaustive literature review.Also cited at Abstract and conditional construction.Also cited at Editorial research route recorded 2026-07-31.Source used to formulate or check the problem record.Source used to assess the problem's recorded status.Packet-linked strongest all-dimension conditional result.Source named by the research packet.
- Christopher A. Fuchs, Michael C. Hoang, and Blake C. Stacey, “The SIC Question: History and State of Play”. Axioms 2017, 6(3), 21. DOI 10.3390/axioms6030021. arXiv:1703.07901 (2017). Abstract and numerical-status survey. ↗preprint · secondary source · arXiv:1703.07901v3 · checked 2026-08-01Source use: original summary.Documents extensive finite-dimensional constructions.
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