ProofAtlas

Complex Hadamard matrices and mutually unbiased bases

Mutually unbiased bases in dimension six

Collaboration beta

Three mutually unbiased bases in six complex dimensions can be constructed explicitly. The question is whether a fourth can ever exist. The packet gives exact tests for a fixed triple and reports that a triple sharing one conjugation cannot extend, but does not force that symmetry on an arbitrary quartet.

M(6)=3,|⟨bj,ck⟩|2=1/6(B≠C).
Known results and sources
Illustration: Three abstract groups each contain six vector nodes, while a fourth outlined group contains a question mark.
Illustration: Three pairwise unbiased bases in C⁶ are known; the existence of any set of four remains open.

Research problem

Exact mathematical statement

Two orthonormal bases B,CB,C of ℂ6\mathbb C^6 are mutually unbiased when every squared cross-overlap is 1/61/6. Let M(6)M(6) be the largest size of a collection of pairwise mutually unbiased orthonormal bases. Determine M(6)M(6), and in particular prove the conjecture

M(6)=3,|⟨bj,ck⟩|2=16for allbj∈B, ck∈C, B≠C.M(6)=3,\qquad |\langle b_j,c_k\rangle|^2=\frac16 \quad\text{for all }b_j\in B,\ c_k\in C,\ B\ne C.

the source gives an explicit tensor-product triple, supplying the lower bound three. It supplies neither an exclusion of every arbitrary quartet nor a counterexample quartet, so the universal upper bound remains open.

Problem infographic

Problem at a glance

Illustration: Two six-by-six squared-overlap tables show an identity matrix within a basis and entries 1/6 between different unbiased bases.
Illustration: Orthonormality and uniform cross-overlaps are separate constraints; the sharp dimension-six maximum is unresolved.

Current mathematical picture

Where work on Mutually unbiased bases in dimension six stands

Open problem

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

Useful failureUse linear unbiasedness, crossing, and positive feasibility alone to exclude an extending basis.

For a fixed triple, the source’s center T₀=E+P/4 has eigenvalues 1,(1/4)²⁰,0¹⁵. It satisfies the stated linear and positive constraints but not idempotency, so positive feasibility is insufficient to decide basis feasibility. Retain the complete idempotency equation, the exact residual-rank-five criterion, or a proved stronger obstruction. A few scalar moments are necessary consequences, not a replacement for the full equations.

Route status · Narrowed route
Main reductionCurrent reduction

Fix any physical triple of mutually unbiased orthonormal bases in C⁶. Form the graph whose vertices are distinct physical rays unbiased to all three reference bases and whose edges mean orthogonality. The source gives a finite common-ray algebra of length at most 246 and proves that a fourth basis exists exactly when the graph has five mutually orthogonal vertices; the sixth projector is I−∑_{r=1}^5p_r. Excluding such a five-clique for every physical triple remains unproved.

Evidence posture · Source-reported route statement · dependencies incomplete
Priority open bridgeExclude arbitrary nonsymmetric quartets, beyond triples with a shared conjugation.Task status · Ready to work on

Work mapped so far

Mutually unbiased bases in dimension six in numbers

1.2kretained lines of mathematical investigation1,154 in the current working snapshot
Argument development
751 · 65%
Explored or eliminated routes
55 · 5%
Computational analysis
102 · 9%
Open obligations
141 · 12%
Definitions and setup
105 · 9%
9selected mapped statements2routes investigated3open questions3contribution-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 byG. ZaunerProposed1991Status checkedSep 7, 2026
Source recordedSep 7, 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

14 selected steps

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

14 selected steps

Scroll horizontally to explore the route

Working route overview for Mutually unbiased bases in dimension sixA selected map of recorded claims, active routes, useful failures, open questions, and their explicit relationships. Search, filter, zoom, or pan within this page.M(6)=3: an explicit triple exists, while exclusion of every fourth basis remains open. — Depends on missing premiseM(6)=3: an explicit tripleexists, while exclusion ofevery…A full real operator family gives an exact rank test — Depends on missing premiseA full real operator familygives an exact rank testAfter three rays, the remaining exact tests are small — Depends on missing premiseAfter three rays, theremaining exact tests aresmallCurrent reduction — Depends on missing premiseCurrent reductionFive orthogonal common rays decide one fixed triple — Depends on missing premiseFive orthogonal common raysdecide one fixed tripleAn explicit tensor construction gives three bases — Depends on missing premiseAn explicit tensorconstruction gives threebasesClosing target — Depends on missing premiseClosing targetOne shared conjugation excludes any fourth basis — Depends on missing premiseOne shared conjugationexcludes any fourth basisThe common-ray algebra keeps multiplicities and physical rays — Depends on missing premiseThe common-ray algebra keepsmultiplicities and physicalraysUse linear unbiasedness, crossing, and positive feasibility alone to exclude an extending basis. — stoppedUse linear unbiasedness,crossing, and positivefeasibility…Assume the residual projection universally lies outside the quadratic direction span. — stoppedAssume the residualprojection universally liesoutside…Exclude arbitrary nonsymmetric quartets, beyond triples with a shared conjugation. — OpenExclude arbitrarynonsymmetric quartets,beyond…Prove strict positivity of the complete three-ray compression tests on all remaining physical inputs. — OpenProve strict positivity ofthe complete three-raycompression…Control every exceptional cubic or operator stratum in an all-input exclusion. — OpenControl every exceptionalcubic or operator stratum inan…
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

2 recorded
Narrowed routeUse linear unbiasedness, crossing, and positive feasibility alone to exclude an extending basis.

For a fixed triple, the source’s center T₀=E+P/4 has eigenvalues 1,(1/4)²⁰,0¹⁵. It satisfies the stated linear and positive constraints but not idempotency, so positive feasibility is insufficient to decide basis feasibility. Retain the complete idempotency equation, the exact residual-rank-five criterion, or a proved stronger obstruction. A few scalar moments are necessary consequences, not a replacement for the full equations.

Route status · Narrowed route
Narrowed routeAssume the residual projection universally lies outside the quadratic direction span.

The source’s EX-5D example has no extension but satisfies P∈span{A²:A∈V}, with an explicit identity in its five directions. The failed implication is not an unresolved lemma. The source retains the stronger transversality condition V∩(span{A²:A∈V}+RP)={0} as a sufficient obstruction, checked on particular examples but not proved universally. Its weak quadratic lift counterexample does not refute a fully degree-closed sum-of-squares approach.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

3 featured tasks
01
Exclude arbitrary nonsymmetric quartets, beyond triples with a shared conjugation.Suggested move: Prove that some three bases in every hypothetical quartet admit one antiunitary involution preserving each individually, or supply a nonsymmetric obstruction covering the remaining triples. Separate conjugations, a basis-permuting antiunitary, and an auxiliary complex structure do not establish that bridge.
Ready to work on
02
Prove strict positivity of the complete three-ray compression tests on all remaining physical inputs.Suggested move: For three selected orthogonal common rays, form the full space K_Q using all reference-frame constraints. Exclude k=3 by −tr([X,Y]²)>0 and k=4 by positivity of the 3×3 commutator Gram determinant, with all degeneracies accounted for. A local D or D+RZ classification alone still permits a completion.
Ready to work on
03
Control every exceptional cubic or operator stratum in an all-input exclusion.Suggested move: Prove that the compatible cubic dimension is always below six, or classify and exclude each rank-jump stratum using the full equations. Preserve projective boundary cases and the conditional seven-basis branch; generic rank and the four zero-kernel examples do not cover the exceptional locus.
Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Sep 7, 2026
Current statusOpen problem

The exact maximum number of pairwise mutually unbiased orthonormal bases in complex dimension six remains unknown. Three bases are known; the conjecture says that four cannot exist. Nonexistence of a complete set of seven is a weaker question than this sharp upper bound.

[1]
External progress

What the literature has established

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

  1. Peer reviewedMcNulty and Weigert establish that a triple of mutually unbiased product bases in dimension six cannot be extended by even one mutually unbiased vector.[2]
  2. Authoritative summaryThe 2026 review traces the explicit no-four conjecture to Zauner's diploma thesis; it distinguishes his 1999 quantum-design restatement.[1]
2 cited sources3 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Current focusMutually unbiased bases in dimension six
Weaker or relaxed formNonexistence of a complete set of seven bases in dimension six

Excluding all quartets would exclude complete sets. The review's reformulations of complete-set existence must not all be relabeled as equivalents of M(6)=3.

[1]
Solved special caseTriples of product bases in C²⊗C³

The 2012 nonextension theorem applies to product triples, not every arbitrary triple in C⁶.

[2]
Related problemComplex Hadamard matrices

The review explains their role in representing pairs and larger collections of unbiased bases; a matrix-family restriction retains its own scope.

[1]

Formalization opportunities

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

  • Formalization targetA proposed formal statement must use complex orthonormal bases and squared overlap 1/6, preserving the difference between three, four, and seven bases.
  • Formalization targetAny extension obstruction needs its exact product, matrix-family, or phase assumptions and a separate bridge to unrestricted quartets.

Detailed research inventory

Claims, milestones, and routes in the current map

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

7 standing statements2 proposed statements3 open questions2 narrowed routes
Statements by mathematical role9 selected mapped statements
  • theorem candidate1 of 91
  • reduction1 of 91
  • lemma4 of 94
  • equivalence3 of 93
Selected mathematical clusters1 mathematical clusters
Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.25 displayed rows · 2 routes included
  • retained route statementM(6)=3: an explicit triple exists, while exclusion of every fourth basis remains open.
  • retained route statementCurrent reductionintermediate
  • retained route statementClosing targetintermediate
  • retained route statementAn explicit tensor construction gives three basesintermediate
  • retained route statementA full real operator family gives an exact rank testintermediate
  • retained route statementThe common-ray algebra keeps multiplicities and physical raysintermediate
  • retained route statementFive orthogonal common rays decide one fixed tripleintermediate
  • retained route statementAfter three rays, the remaining exact tests are smallintermediate
  • retained route statementOne shared conjugation excludes any fourth basisintermediate
  • 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
  • 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 failureUse linear unbiasedness, crossing, and positive feasibility alone to exclude an extending basis.reported failure
  • Useful failureAssume the residual projection universally lies outside the quadratic direction span.reported failure
  • Research targetExclude arbitrary nonsymmetric quartets, beyond triples with a shared conjugation.open
  • Research targetProve strict positivity of the complete three-ray compression tests on all remaining physical inputs.open
  • Research targetControl every exceptional cubic or operator stratum in an all-input exclusion.open
  • ComputationThe source reports exact example triples with direction dimensions 12, 5, 2, and 4 and zero compatible cubic kernels. A separate cubic linear-algebra route checks rank 1771 on each at prime 241 while reusing the input constructors. Support-three counts are nested: controlling 1508 excluded/792 surviving, first-audited 1604/696, and working 1706/594.The four examples do not classify all triples, and the different cubic route is not an independent reconstruction of their inputs. Nested support counts must not be added. Pair-six independent promotion, missing producers, and unfinished interval workloads remain separate gaps; no current all-parameter certificate is claimed. · reported unreproduced
  • Narrowed routeUse linear unbiasedness, crossing, and positive feasibility alone to exclude an extending basis.For a fixed triple, the source’s center T₀=E+P/4 has eigenvalues 1,(1/4)²⁰,0¹⁵. It satisfies the stated linear and positive constraints but not idempotency, so positive feasibility is insufficient to decide basis feasibility. Retain the complete idempotency equation, the exact residual-rank-five criterion, or a proved stronger obstruction. A few scalar moments are necessary consequences, not a replacement for the full equations.
  • Narrowed routeAssume the residual projection universally lies outside the quadratic direction span.The source’s EX-5D example has no extension but satisfies P∈span{A²:A∈V}, with an explicit identity in its five directions. The failed implication is not an unresolved lemma. The source retains the stronger transversality condition V∩(span{A²:A∈V}+RP)={0} as a sufficient obstruction, checked on particular examples but not proved universally. Its weak quadratic lift counterexample does not refute a fully degree-closed sum-of-squares approach.
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 bridgeExclude arbitrary nonsymmetric quartets, beyond triples with a shared conjugation.

2 approaches have 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 pointExclude arbitrary nonsymmetric quartets, beyond triples with a shared conjugation.

Mutually unbiased bases in dimension six · ready to start

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Research contextPrepared context for any AI agent

Three mutually unbiased bases in six complex dimensions can be constructed explicitly. The question is whether a fourth can ever exist. The current work gives exact tests for a fixed triple and reports that a triple sharing one conjugation cannot extend, but does not force that symmetry on an arbitrary quartet.

  • Exact question and boundaries
  • Current routes and known obstacles
  • What a useful result should report
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Sources and references2 cited works · next context review by Dec 7, 2026

The mathematical context was checked on Sep 7, 2026. Status can be refreshed sooner after a material result or claim.

  1. 1
    Mutually Unbiased Bases in Composite Dimensions – A Reviewsurvey or monograph · Daniel McNulty, Stefan Weigert · Quantum 10, 2051 · 2026-04-01 · ARXIV 2410.23997 · DOI 10.22331/q-2026-04-01-2051 · accessed Sep 7, 2026
  2. 2
    On the Impossibility to Extend Triples of Mutually Unbiased Product Bases in Dimension Sixpeer reviewed result · Daniel McNulty, Stefan Weigert · International Journal of Quantum Information 10, 1250056 · 2012 · ARXIV 1203.6887 · DOI 10.1142/S0219749912500566 · accessed Sep 7, 2026

Important qualifications

  • This is a bounded external-context collection, not proof checking, an independent mathematical review, legal clearance, or publication authority.
  • No source programs, datasets, or certificates were executed. A linked resource does not imply reproduction.
  • No exact checked formalization was confirmed in the scoped search; an empty list does not establish nonexistence.
  • Unassigned history and recognition fields mean this collection did not establish them. Proposed formal prerequisites are planning tasks, not an exhaustive library inventory.
  • The 1991 date and attribution come from the 2026 review's historical account, not direct inspection of Zauner's thesis.
  • The review appeared in 2026 but describes its survey snapshot as summer 2024; this collection also searched current sources and does not assert an exhaustive September 2026 literature audit.
  • No exact machine-checked statement or proof was inspected. Search hits claiming formal targets were not promoted to formalization resources.

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