# P46: Zauner's conjecture

- ID: `P46`
- Reference: `zauners-conjecture`
- Page: https://theoremdb.org/statements/P46
- Record maturity: Reviewed problem with recorded work

## Problem

For every integer \(d\ge2\), there exist \(d^2\) unit vectors in \(\mathbb{C}^d\) whose pairwise squared inner-product magnitudes are all \(1/(d+1)\).

### Context

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

### Problem setup

- **Definition (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 (Weyl-Heisenberg covariance).** Weyl-Heisenberg covariance means that all d^2 lines arise from one fiducial vector under the discrete displacement operators.
- **Remark.** The conjecture links optimal quantum measurements with equiangular lines, number fields, and special values of analytic functions.

### What 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.

## Status

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](#reference-1) [2](#reference-2)

## Work

### Evidence for the current status

**Claim 1 (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.

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.

### Background and intake notes

- 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 2026-07-22. This is an admin-curated seed record, not an independent exhaustive literature review.
- The formulation and status were checked against the cited paper and a July 2026 Journal of Number Theory article on 2026-07-22.
- Exact and numerical SICs are known in many dimensions. Conditional constructions using Stark-type conjectures do not settle unconditional existence.

### Open directions

- **Route 1** (reported): 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](#reference-1)

### Computational notes

- High-precision numerical fiducials provide evidence in particular dimensions but require exact certification and do not cover all dimensions.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `zauners-conjecture`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>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 https://arxiv.org/abs/2501.03970
   - Also cited at Abstract and conditional construction
   - Also cited at Editorial research route recorded 2026-07-31
   - preprint; primary source; arXiv:2501.03970, checked 2026-07-31; checked 2026-07-31
   - Source 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.
   - 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.
2. <a id="reference-2"></a>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 https://arxiv.org/abs/1703.07901
   - preprint; secondary source; arXiv:1703.07901v3; checked 2026-08-01
   - Source use: original_summary
   - Documents extensive finite-dimensional constructions.