finite.works
Results · Quantum correlations

How much dimension does an ε-approximation to a non-attained quantum correlation cost? A Diophantine answer for K5

Seth Douglas · version 1.0.2 · Zenodo, · doi:10.5281/zenodo.22944181

In plain wordsIn a standard example where finite quantum systems approach a correlation they never reach, coming within ε of it takes dimension of order the fourth root of 1/ε, an exponent fixed by how well the square root of 5 is approximated by fractions.

The result

As the paper states it, in briefThe dimension cost of approximating a non-attained correlation for K5 follows a Diophantine law. Version 1.0.2 shows that an open lemma conjectured in earlier versions is false and settles the drifted case.

Questionson the index

How this was checked

Independent computation
Three engines check the exact claims of Sections 2 to 6 and Lemmas 4.5 and 4.6.
Lean
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Peer review
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Novelty and credit

Novelty
New, with named precedent: Coladangelo’s embezzlement proof of non-closure (arXiv:1904.02350) carries a one-sided dimension bound for its witness. The structure used is credited to Kruglyak, Rabanovich and Samoĭlenko and to Dykema, Paulsen and Prakash. The paper labels its priority provisional: no evidence found, not proved absent.
Correction
Version 1.0.2 shows that an open lemma of earlier versions is false. Corrections

Versionsthree, each with its own DOI

Every archived version, newest first. Dates are Zenodo publication dates. All versions: 10.5281/zenodo.21876896, which always resolves to the newest.
VersionDateDOINotes
1.0.210.5281/zenodo.22944181Titled “How much dimension does an epsilon-approximation to a non-attained quantum correlation cost? A Diophantine answer for K_5”. Correction: An open lemma of versions 1.0.0 and 1.0.1 conjectured that marginal drift δ costs δ^−1/2. It is false: an explicit two-block carrier has deficit exactly zero with drifted marginals. Version 1.0.2 proves the cost Θ(min(ε^−1/4, δ^−1/4)).
1.0.110.5281/zenodo.22099141Titled “How much dimension does an epsilon-approximation to a non-attained quantum correlation cost? A Diophantine answer for K_5”. Correction: Credit: version 1.0.0 said that Coladangelo’s embezzlement proof of non-closure (arXiv:1904.02350) comes with no rate; its abstract states an explicit dimension lower bound, 2^Ω(ε^−1/8). The priority statement is now scoped to the K5 witness. The theorem and constants are unchanged.
1.0.010.5281/zenodo.21876899Titled “How much dimension does an epsilon-approximation to a non-attained quantum correlation cost? A Diophantine answer for K_5”.

Paper, code and links

Paper

This release has no PDF: the paper is the Markdown file paper/PAPER-K5.md at the archived commit.

Corrections

  1. Version 1.0.2. An open lemma of versions 1.0.0 and 1.0.1 conjectured that marginal drift δ costs δ−1/2. It is false: an explicit two-block carrier has deficit exactly zero with drifted marginals. Version 1.0.2 proves the cost Θ(min(ε−1/4, δ−1/4)).doi:10.5281/zenodo.22944181
  2. Version 1.0.1. Credit: version 1.0.0 said that Coladangelo’s embezzlement proof of non-closure (arXiv:1904.02350) comes with no rate; its abstract states an explicit dimension lower bound, 2Ω(ε−1/8). The priority statement is now scoped to the K5 witness. The theorem and constants are unchanged.doi:10.5281/zenodo.22099141

All corrections, and what counts as one.

Cite

@misc{douglas2026k5,
  author  = {Douglas, Seth},
  title   = {How much dimension does an ε-approximation to a non-attained quantum correlation cost? {A} {D}iophantine answer for {K5}},
  year    = {2026},
  version = {1.0.2},
  doi     = {10.5281/zenodo.22944181},
  url     = {https://doi.org/10.5281/zenodo.22944181}
}

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