bibkey: mcnultyweigert2024mutually authors: Daniel McNulty, Stefan Weigert year: 2024 title: Mutually Unbiased Bases in Composite Dimensions – A Review doi: 10.48550/arXiv.2410.23997 claim: Problem 10.2 states the nonexistence question for four mutually unbiased bases in dimension six; separately, the review’s fourteen equivalent formulations concern complete MUB sets (Fin 7 in dimension six), not the Fin 4 problem. strata_touched:

• D5/S3/Quantum/Tomography/RankOneContextCommutator
• D5/S3/Quantum/Tomography/CompleteContextTomography
• D5/S3/Quantum/Tomography/ComplementaryContextProbabilityPythagoras
• D5/S3/QuantumBounds/Designs/CollisionConservation license: citation-only triage: anchor

Mutually Unbiased Bases in Composite Dimensions – A Review

McNulty and Weigert review the construction and nonexistence questions for mutually unbiased bases in composite dimensions. Problem 10.2 directly asks to show that no set of four MU bases exists when d = 6; it is the main source anchor for Problems/mub-six-fourth-basis.md. Problem 10.1 is the complete-set comparison, asking for nonexistence of seven MU bases in dimension six. Conjecture 2.1 is retained as Zauner’s original k = 4 affine quantum-design formulation of the fourth-basis problem.

The review’s abstract also reports fourteen mathematically equivalent formulations of the complete-set existence problem. In dimension six those formulations concern Fin 7, not Fin 4. This note does not transfer those equivalences into the repository and does not treat any numerical obstruction as a proof of nonexistence.

Search log

• 2026-08-22: The orchestrator queried the arXiv Atom API with id_list=2410.23997. HTTP 200 returned one entry whose title and authors matched this note. The entry was published 2024-10-31, updated 2026-03-26, and assigned primary category quant-ph; it also reported journal DOI 10.22331/q-2026-04-01-2051. The same probe returned one entry for positive control 2002.03233 and zero entries for negative control 9999.99999.
• 2026-08-22: The orchestrator issued HEAD https://doi.org/10.48550/arXiv.2410.23997. It returned HTTP 302 to https://arxiv.org/abs/2410.23997; negative control 10.48550/arXiv.9999.99999 returned HTTP 404.
• 2026-08-22: The orchestrator fetched the raw arXiv HTML for version 2 with curl -sL https://arxiv.org/html/2410.23997v2; the response was HTTP 200 and 2,461,445 bytes. grep -oE 'Problem 10\.[0-9]+' found all fifteen labels from Problem 10.1 through Problem 10.15. The source reads: “Problem 10.1. Show that no complete set of seven MU bases exists when d = 6.” and “Problem 10.2. Show that no set of four MU bases exists when d = 6.” In the same fetch, Conjecture 1.1 occurred four times and Conjecture 2.1 once.
• 2026-08-22: The orchestrator’s full-tree fixed-string search found zero D5 hits for MutuallyUnbiased, mutually_unbiased, MutuallyUnbiasedBases, ComplexHadamard, complexHadamard, and HadamardMatrix; RankOneContext supplied the positive control. The implementation worker reproduced these zero counts and the positive-control hit in the current checkout.

Verified locator

• arXiv: https://arxiv.org/abs/2410.23997
• arXiv DOI: https://doi.org/10.48550/arXiv.2410.23997
• arXiv HTML v2: https://arxiv.org/html/2410.23997v2
• Journal DOI, not the canonical DOI of this note: https://doi.org/10.22331/q-2026-04-01-2051