The price of precision: usable entanglement in nonlocal games and the geometry of groups

As the paper states it, in briefFor the game classes treated, the entanglement demand near perfect play diverges exactly when no finite-dimensional strategy is perfect; bounded demand is undecidable among games of value one; and the geometry of groups gives logarithmic, almost inverse-loss and intermediate laws.

• Answered When does near-perfect play need unbounded entanglement, and can one tell?
• Open Is every positive usable-entanglement coefficient of a finite rational game limit-computable? left open

Blind audits

Repeated rounds of blind audit for version 1.0, the final round with no serious findings; four further rounds for version 1.1.

Lean

None yet.

Peer review

Not yet.

Every check here is internal. How we check results.

Label checked 30 October 2026.

Novelty

New, with named precedent: Mančinska and Vidick gave a game, with answers of unbounded length, whose optimal value no finite amount of entanglement attains. Every imported result is credited at its use, among them Slofstra’s and those of Fanizza, Kroell, Mehta, Paddock, Rochette, Slofstra and Zhao. Bounded literature searches found no exact preemption; the paper makes no claim of being first.

PDF

papers/price-of-precision-v1.1.0.pdf, version 1.1.0, 2,034,112 bytes, 160 pages.

SHA-256 85d1768202a8887885929230135192fe6eba304efc8d367f2b4149710bfa936d

DOI

10.5281/zenodo.23753349; all versions 10.5281/zenodo.23697345

Code

github.com/Apsiape/sharp-entanglement-bounds

None so far.

All corrections, and what counts as one.

@misc{douglas2026priceofprecision,
  author  = {Douglas, Seth and Mghirbi, Nidhal},
  title   = {The price of precision: usable entanglement in nonlocal games and the geometry of groups},
  year    = {2026},
  version = {1.1.0},
  doi     = {10.5281/zenodo.23753349},
  url     = {https://doi.org/10.5281/zenodo.23753349}
}

price-of-precision.bibAll papers as BibTeX