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Quantum information · frame theory · Weyl–Heisenberg covariance

Zauner’s Conjecture and SIC-POVM Existence

Collaboration beta

Does every complex dimension at least two admit a Weyl–Heisenberg covariant SIC fiducial, preferably with canonical order-three Zauner symmetry?

|⟨ψ,Dpψ⟩|2=1d+1(p≠0)
Known results and sources
A central quantum-state ray generates a symmetric constellation of equally spaced projective rays around a phase-space grid.
Zauner’s conjecture asks whether every finite complex dimension admits a Weyl–Heisenberg orbit with exactly equal nontrivial pairwise overlaps.

Research problem

Exact mathematical statement

Let d≥2d\ge2, let Hd=ℂd\mathcal H_d=\mathbb C^d, and choose Weyl–Heisenberg displacement operators {Dp:p∈ℤd2}\{D_p:p\in\mathbb Z_d^2\} in any consistent standard phase convention. A unit vector ψ∈Hd\psi\in\mathcal H_d is a Weyl–Heisenberg SIC fiducial when

|⟨ψ,Dpψ⟩|2=1,&p=0,1d+1,&p≠0.|\langle\psi,D_p\psi\rangle|^2= \begin{cases} 1,&p=0,\\ \frac1{d+1},&p\ne0. \end{cases}

Its orbit projectors then form a SIC-POVM after normalization by 1/d1/d. The strong Zauner target is:

For every d≥2d\ge2, there exists a Weyl–Heisenberg SIC fiducial that is an eigenvector of a canonical order-three Clifford unitary UZU_Z.

The weaker target omits the Zauner eigenvector condition. The governing source states that the universal conjecture is not solved.

Problem infographic

Problem at a glance

A fiducial state is displaced across a finite phase-space grid into a symmetric ray constellation whose equal overlaps remain an all-dimensional question.
Weyl–Heisenberg displacements of one fiducial should produce d² projective rays with the same nontrivial overlap; the strong target also asks for order-three Zauner symmetry.

Current mathematical picture

Where work on Zauner’s Conjecture and SIC-POVM Existence stands

Open conjecture

Selected route highlights from the mathematical source. This is not yet a complete mathematical inventory.

Useful failureLocal landscape and support-only arguments as universal existence proofs

The source records cubic or Alltop non-SIC local minima and notes that generic full-support Zauner eigenvectors can still have many distinct nonzero overlap magnitudes. An exact global minimum certificate in dimension 11, a representation-theoretic no-missing-line identity, a rank-one-aware moment relaxation, or an all-dimensional Weyl-compatible construction remain source-proposed paths.

Route status · Narrowed route
Main reductionConditional biangular classification

In the four certified primes, any canonical-Zauner Weyl-biangular state would be a SIC, but existence of such a state is not proved.

Evidence posture · Source-reported route statement · dependencies incomplete
Priority open bridgeProve the dimension-11 Zauner-eigenspace minimum reaches the SIC value.Task status · Ready to work on
Research-record correctionResearch-record correction

We corrected the cited passages. We removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Reader-facing record corrected; mathematics unchanged

Work mapped so far

Zauner’s Conjecture and SIC-POVM Existence in numbers

1.4kretained lines of mathematical investigation1,408 in the current working snapshot
Argument development
1,162 · 83%
Explored or eliminated routes
43 · 3%
Computational analysis
56 · 4%
Open obligations
55 · 4%
Definitions and setup
92 · 7%
7selected mapped statements1routes investigated4open questions4contribution-ready tasks
How this is measured

This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.

Proposed byGerhard ZaunerProposed1999Status checkedAug 15, 2026
Source recordedAug 15, 2026

Argument map and routes

How the current approaches connect

Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.

Visible working map

Research route map

12 selected steps

Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.

12 selected steps

Scroll horizontally to explore the route

Working route overview for Zauner’s Conjecture and SIC-POVM ExistenceA selected map of recorded claims, active routes, useful failures, open questions, and their explicit relationships. Search, filter, zoom, or pan within this page.Every finite complex dimension should admit a Weyl–Heisenberg SIC — Depends on missing premiseEvery finite complexdimension should admit aWeyl–Heisenberg…Conditional biangular classification — Depends on missing premiseConditional biangularclassificationCurrent reduction — Depends on missing premiseCurrent reductionFourth-moment characterization — Depends on missing premiseFourth-momentcharacterizationWeyl–Heisenberg SIC condition — Depends on missing premiseWeyl–Heisenberg SICconditionClosing target — Depends on missing premiseClosing targetFour-prime no-missing-line certificates — Depends on missing premiseFour-prime no-missing-linecertificatesLocal landscape and support-only arguments as universal existence proofs — stoppedLocal landscape andsupport-only arguments asuniversal…Prove the dimension-11 Zauner-eigenspace minimum reaches the SIC value. — OpenProve the dimension-11Zauner-eigenspace minimumreaches…Extract a dimension-uniform analytic no-missing-line identity. — OpenExtract a dimension-uniformanalytic no-missing-lineidentity.Construct the all-dimensional bridge. — OpenConstruct theall-dimensional bridge.Global biangularity or attainment gap — OpenGlobal biangularity orattainment gap
Working claimActive routeOpen, active, or blocked questionUseful failure

Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.

Explored alternatives

Other routes

1 recorded
Narrowed routeLocal landscape and support-only arguments as universal existence proofs

The source records cubic or Alltop non-SIC local minima and notes that generic full-support Zauner eigenvectors can still have many distinct nonzero overlap magnitudes. An exact global minimum certificate in dimension 11, a representation-theoretic no-missing-line identity, a rank-one-aware moment relaxation, or an all-dimensional Weyl-compatible construction remain source-proposed paths.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

4 featured tasks
01
Prove the dimension-11 Zauner-eigenspace minimum reaches the SIC value.Suggested move: Seek an exact SOS or algebraic global certificate for the fourth-moment deficit on the stated eigenspace, not another numerical fiducial search.
Ready to work on
02
Extract a dimension-uniform analytic no-missing-line identity.Suggested move: Decompose the finite SOS Gram forms under the residual Clifford and Zauner symmetry and express the quartic as canonical squared covariants controlled by exact exponential-sum bounds.
Ready to work on
03
Global biangularity or attainment gap

The decisive missing implication is to force a global or Zauner-eigenspace minimizer to attain the SIC energy or have the biangular structure classified by the certificates.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.
Ready to work on
04
Construct the all-dimensional bridge.Suggested move: Test tensor, prime-power, and Chinese-remainder constructions that preserve Weyl covariance, rank-one purity, and exact pairwise overlaps in composite and even dimensions.
Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 15, 2026
Current statusOpen conjecture

Existence of SICs in every finite dimension remains open, as does the stronger Weyl–Heisenberg/Zauner-symmetric formulation. Extensive finite numerical and exact constructions provide evidence but do not establish the universal statement.

[4][3]
External progress

What the literature has established

Selected external milestones in reverse chronological order, with their evidence posture.

  1. Peer reviewedYakymenko related canonical order-three unitaries to triangle-group representations while explicitly retaining all-dimensional existence as open.[4]
  2. Computational resultScott extended the numerical list with one Zauner-symmetric solution in every dimension through 121 and selected higher dimensions.[3]
  3. Peer reviewedScott and Grassl reported numerical SIC solutions in every dimension through 67 and a putatively complete covariant list through 50.[2]
  4. Historical sourceZauner's dissertation formulated all-dimensional Weyl–Heisenberg SIC existence with an additional order-three symmetry.[1]
4 cited sources3 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Current focusZauner's conjecture for Weyl–Heisenberg SIC-POVM existence
Weaker or relaxed formUniversal SIC existence

Universal SIC existence without a prescribed Weyl–Heisenberg orbit or Zauner eigenvector is weaker than the strong source statement but is also open.

[4]
Solved special caseFinite-dimensional numerical SIC constructions

Numerical Weyl–Heisenberg fiducials exist across long finite ranges and selected higher dimensions; finite computational coverage does not imply existence in every dimension.

[2][3]
Related problemTriangle-group representation structure

Canonical order-three Clifford symmetries arise through projective representations of the triangle group (3,3,3).

[4]

Formal and computational footholds

Existing statements, libraries, computations, and datasets that can shorten the next serious attempt.

  • dataset · source linked; not reproduced by ProofAtlasScott–Grassl numerical SIC solution lists

    Primary studies report numerical Weyl–Heisenberg fiducials through substantial finite dimension ranges and selected higher dimensions; these are computational data, not universal proof.

    [2][3]

Formalization opportunities

Lean work can make these reusable foundations precise without being presented as a proof of the core problem.

  • Formalization targetNo all-dimensional formal proof or construction of SIC-POVM existence was identified.
  • Formalization targetFormal infrastructure would need finite-dimensional complex Hilbert spaces, Weyl–Heisenberg and Clifford groups, equiangular tight frames, and exact algebraic or analytic existence arguments.

Research-record corrections

What changed in the research record

These notes describe corrections to cited passages, highlighted tasks, or connections between claims. The mathematical claims and their status did not change.

Research-record correctionWe corrected the cited passages. We removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details

The initial argument structure appears separately. Uploads, model runs, and presentation changes do not count as mathematical updates.

Detailed research inventory

Claims, milestones, and routes in the current map

This view highlights the mathematical statements most useful for following the current route.

5 standing statements2 proposed statements4 open questions1 narrowed routes
Statements by mathematical role7 selected mapped statements
  • theorem candidate1 of 71
  • reduction2 of 72
  • lemma2 of 72
  • equivalence2 of 72
Selected mathematical clusters1 mathematical clusters
Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.20 displayed rows · 1 route included
  • retained route statementEvery finite complex dimension should admit a Weyl–Heisenberg SIC
  • retained route statementCurrent reductionintermediate
  • retained route statementClosing targetintermediate
  • retained route statementWeyl–Heisenberg SIC conditionintermediate
  • retained route statementFourth-moment characterizationintermediate
  • retained route statementFour-prime no-missing-line certificatesintermediate
  • retained route statementConditional biangular classificationintermediate
  • Recorded relationshipThe source reports this as a route toward the conjecture; missing or unaudited premises remain and the reduction does not itself prove the target.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • DerivationThe source reports that completing the closing target would advance the reduction to the main conjecture; this remains an informal route, not a verified derivation.proposed
  • Useful failureLocal landscape and support-only arguments as universal existence proofsreported failure
  • Research targetProve the dimension-11 Zauner-eigenspace minimum reaches the SIC value.open
  • Research targetExtract a dimension-uniform analytic no-missing-line identity.open
  • Research targetConstruct the all-dimensional bridge.open
  • Research targetGlobal biangularity or attainment gapopen
  • Research targetAll-dimensional constructionsuperseded
  • Narrowed routeLocal landscape and support-only arguments as universal existence proofsThe source records cubic or Alltop non-SIC local minima and notes that generic full-support Zauner eigenvectors can still have many distinct nonzero overlap magnitudes. An exact global minimum certificate in dimension 11, a representation-theoretic no-missing-line identity, a rank-one-aware moment relaxation, or an all-dimensional Weyl-compatible construction remain source-proposed paths.
How to interpret these counts

A statement may be a lemma, conditional reduction, special case, documented limitation, or open target. These counts describe the work's structure; they do not estimate distance to a proof.

Research outlook

Conditions that would advance the current route

Priority open bridgeProve the dimension-11 Zauner-eigenspace minimum reaches the SIC value.

1 approach has already been tested and narrowed. The task above is the current priority within the larger open route.

Evidence needed nextConcrete conditions for progress

A result can change the outlook by closing the bridge, narrowing its scope, or showing that the route cannot work.

  • Supply a complete argument with every imported premise identified.
  • Survive an independent attempt to falsify the proposed step.

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Prepared starting pointProve the dimension-11 Zauner-eigenspace minimum reaches the SIC value.

Zauner’s Conjecture and SIC-POVM Existence · ready to start

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Research contextPrepared context for any AI agent

Does every complex dimension at least two admit a Weyl–Heisenberg covariant SIC fiducial, preferably with canonical order-three Zauner symmetry?

  • Exact question and boundaries
  • Current routes and known obstacles
  • What a useful result should report
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Sources and references4 cited works · next context review by Nov 14, 2026

The mathematical context was checked on Aug 15, 2026. Status can be refreshed sooner after a material result or claim.

  1. 1
    Quantum Designs: Foundations of a Noncommutative Design Theoryoriginal source · Gerhard Zauner · University of Vienna / author-hosted translation · 1999; English translation 2010 · accessed Aug 15, 2026
  2. 2
    SIC-POVMs: A new computer studypeer reviewed result · A. J. Scott, M. Grassl · Journal of Mathematical Physics · 2010 · ARXIV 0910.5784 · DOI 10.1063/1.3374022 · accessed Aug 15, 2026
  3. 3
    SICs: Extending the list of solutionspreprint · A. J. Scott · arXiv · 2017-03-11 · ARXIV 1703.03993 · accessed Aug 15, 2026
  4. 4
    SICs and the Triangle Group (3,3,3)peer reviewed result · Danylo Yakymenko · SIGMA · 2024-05-29 · ARXIV 2312.13400 · DOI 10.3842/SIGMA.2024.044 · accessed Aug 15, 2026

Important qualifications

  • Scoped to Zauner's thesis translation, two primary computational studies, and a recent peer-reviewed source explicitly describing the conjecture as open; the SIC solution census was not exhaustively refreshed.
  • Numerical or finite exact solutions are kept separate from an all-dimensional existence proof.
  • The weaker universal SIC-existence question is kept distinct from the stronger Weyl–Heisenberg and canonical order-three symmetry formulation.

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