[#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 \(1/(d+1)\).

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Definitions and notation

1Context

The conjecture links optimal quantum measurements with equiangular lines, number fields, and special values of analytic functions.

2Problem setup

Remark 1. The corresponding rank-one operators form a symmetric informationally complete positive operator-valued measure, or SIC-POVM.

Definition 1 (Weyl-Heisenberg covariance). Weyl-Heisenberg covariance means that all d^2 lines arise from one fiducial vector under the discrete displacement operators.

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

Saved packet · July 31, 2026

What counts as a solution

Saved packet 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]

1Packet records

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Record	Kind	Assessment
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.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 1 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] Relevance to this problem For Zauner's conjecture, this successor gives readable dated status prose and the exact remaining research boundary. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1, Reference 2 Details The packet's cited sources and equivalent formulations were checked in the dated review recorded below. 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. Open full record →
Construct a Weyl-Heisenberg covariant SIC in every dimension d at least 2, or give a dimension and prove that no such SIC exists.	Trace	Outcome: Open Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 1 Construct a Weyl-Heisenberg covariant SIC in every dimension d at least 2, or give a dimension and prove that no such SIC exists.^[1] Relevance to this problem For Zauner's conjecture, record zauners-conjecture-attempt-resolution-route (“Resolve the stated acceptance condition”) documents a concrete method, search boundary, or failed route. The record states: Construct a Weyl-Heisenberg covariant SIC in every dimension d at least 2, or give a dimension and prove that no such SIC exists. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded Sources Reference 1 What happened Target the displayed statement directly. Construct a Weyl-Heisenberg covariant SIC in every dimension d at least 2, or give a dimension and prove that no such SIC exists. Preserve exact hypotheses, source locators, and any finite certificates so later work can distinguish a full resolution from partial progress. Open full record →
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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

Uniform Exponential Quantum Parallel Repetitionquantum information
Exponential instability at the real phase-retrieval thresholdframe theory
Lehmer's Mahler measure problemalgebraic number theory

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Plain text

“Zauner's conjecture.” TheoremDB. P46. Problem statement; statement text SHA-256 621da661f72a5976a3d49b62d148bf8377a8900e9bc7f1598c99f96fae55bf79. https://theoremdb.org/statement/?ref=P46

BibTeX

@misc{theoremdb-problem-621da661f72a5976a3d49b62d148bf8377a8900e9bc7f1598c99f96fae55bf79,
  title = {{Zauner's conjecture}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 621da661f72a5976a3d49b62d148bf8377a8900e9bc7f1598c99f96fae55bf79},
  url = {https://theoremdb.org/statement/?ref=P46}
}

Plain text: Built Markdown snapshot

This problem includes 2 records joined by 2 typed links, sourced from arxiv.org^[1], current as of July 31, 2026.

1Lean verification

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.

An original CC0 restatement prepared by TheoremDB maintainers.

[#P46] Zauner's conjecture

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