TBNQuantum
A trace costs one quantifier: the exact complexity of perfect play in nonlocal games
Publication: quant-ph updates on arXiv.org
Author: Seth Douglas, Nidhal Mghirbi
Source published: 10/11/2026, 05:00:00 UTC
Briefing created: 10/11/2026, 10:06:47 UTC
Based on a feed excerpt; full article content has not been acquired. Source evidence retrieved 10/11/2026, 08:43:25 UTC.
Briefings
A new proof establishes that perfect play in nonlocal games is $\Sigma^0_2$-complete, strictly harder than previous models. The evidence shows this complexity holds even for games of value one, requiring a trace to cost one existential quantifier. Key findings include the hardness of perfect normal strategies in commuting models and finite entanglement entropy scenarios, all verified in Lean. The paper constructs finite channels with zero-error codes needing infinite entanglement entropy, and proves a strict ladder between finite rank and finite entropy conditions.
Key points
- The existence of a perfect normal spatial strategy is $\Sigma^0_2$-complete in nonlocal games of value one, making it undecidable even with an oracle for the halting problem.
Context
This research addresses the computational complexity of perfect play in nonlocal games, specifically proving $\Sigma^0_2$-completeness for finite games of value one. The findings clarify the hierarchy between finite-dimensional and commuting-operator models, with commuting strategies being $\Sigma^0_1$-complete. The work is formally verified in Lean and provides a strict ladder between finite rank and finite entropy conditions.
Limitations
The evidence is from an arXiv preprint (arXiv:2611.10481v1) and does not include peer-reviewed publication status or final experimental validation. The complexity results are theoretical proofs, and the specific computational cost of implementing these strategies remains unverified.
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