Quantum Advantage in Threefold Repetition of Feige’s Game

Research-attempt provenance

Definitions and mathematical statement

Shared notation.

Standard number systems are denoted by $\mathbb N,\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$; index ranges, computational models and specialized notation follow the cited problem source and the definitions below.

A game's classical value maximizes winning probability using classical strategies; its quantum value allows the specified entangled strategies. Parallel repetition plays three independent question instances and requires the source's combined winning condition.

An $n$-qubit pure state is a unit vector in $(\mathbb C^2)^{\otimes n}$, considered up to global phase. A quantum algorithm applies permitted quantum operations and measurements; complexity depends on the gate and input-access model.

Statement recorded in the supplied source.

Does the quantum value exceed the classical value for the threefold parallel repetition of Feige’s game? [S2]

Scope qualification.

Feige's game distribution, answer spaces, quantum correlation model, and joint acceptance predicate are not reproduced in the supplied question.

Source status.

The supplied catalogue labels this record open. Its source-status metadata was checked against the supplied snapshot on 2026-10-11. This import does not independently establish current open status or mathematical correctness. The catalogue literature assessment is dated 2026-08-22.

Short English statement

Determine whether entanglement improves the three-repeat value of the cited game.

Sources

• S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 30006570 (OWR-14299906-007). Catalogue curation and metadata: CC BY 4.0. The supplied source question is retained; definitions and the English summary are editorial additions. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Noncommutative Harmonic Analysis and Quantum Information $2026$, Oberwolfach Reports, DOI 10.4171/owr/2026/4, pp. 217--275 https://doi.org/10.4171/owr/2026/4.
• S3 Schmidt--Storgaard--Walter--Zhao, Guess your neighbor's input https://arxiv.org/abs/2510.08484. Underlying Oberwolfach Reports material retains its source-specific licence, including CC BY-SA 4.0 where applicable; see THIRD_PARTY_NOTICES.md.

Preparation and review.

Original question attributed to the cited source and UnsolvedMath contributors. Codex (GPT-6; exact serving revision not exposed) assisted with definitions, English summaries, source inspection and formatting. No new human mathematical authorship or independent proof verification is asserted. Canonical ID 30006570 and source code OWR-14299906-007 checked against the complete supplied registry and existing repository statements; known flagged repetitions and obvious identical conjectures excluded. Full mathematical-equivalence/literature review has not been performed. Statement preparation and final review followed contribution prompts C and D; automated checks verify structure and identity, not open status.

Literature and context check

The primary report, printed pp. 239--241, and the consulted paper arXiv:2510.08484v3, especially its game definition and Section 6.3, specify the missing predicate. They report classical value $5/16$ and nonsignalling value $1/3$ for three repetitions, leaving the quantum comparison open in those sources. Those upper bounds are external results, not re-established here. We verify a reported classical strategy and test restricted improvement routes; no current-status claim beyond these consulted sources is made.

Research attempt: Testing product and one-sided quantum improvements

In one round, independent uniform bits $x,y$ are sent to Alice and Bob. Answers are $0,1,\perp$, and they win exactly for $(a,b)=(\perp,x)$ or $(a,b)=(y,\perp)$. Three repetitions require all coordinates to win. For the following known strategy from the cited paper, write inputs as $(x_1,x_2,x_3),(y_1,y_2,y_3)$: \[ a_1=\perp,\quad a_2=x_1\vee x_3,\quad a_3=\begin{cases}\perp&x_1=0,\\1&x_1=1,\end{cases} \] \[ b_1=y_2\wedge y_3,\quad b_2=\perp,\quad b_3=\begin{cases}y_2&y_2\wedge y_3=0,\\\perp&y_2\wedge y_3=1.\end{cases} \] The first two rounds require $x_1=y_2\wedge y_3$ and $y_2=x_1\vee x_3$. These conditions also guarantee the third round, by separating $x_1=0$ and $x_1=1$. Among the eight triples $(y_2,y_3,x_3)$, five satisfy $y_2=(y_2\wedge y_3)\vee x_3$; the independent required value of $x_1$ contributes a factor $1/2$. The value is therefore $5/16$.

The direct product of the source's optimal two-round strategy and optimal one-round quantum strategy has value $(1/2)(9/16)=9/32$, below $5/16$ by $1/32$. Three independent optimal one-round strategies give $(9/16)^3=729/4096<5/16$. Shared-randomness mixtures of these strategies and the displayed classical strategy have scores equal to the corresponding convex combinations, and hence cannot exceed $5/16$. Thus neither independent tensoring nor convex mixing of these candidates closes the gap.

Why changing only the quantum resources on one commuting side cannot help.

Suppose Alice's finite-dimensional POVM effects for all her input triples commute pairwise. They can be simultaneously diagonalized: diagonalize one Hermitian effect, observe that all the others preserve each eigenspace by commutation, and repeat on those eigenspaces for the finite collection. In a common orthonormal basis indexed by $\ell$, write their diagonal entries as $a_{\mathbf a}^{\mathbf x}(\ell)\geq0$, summing to one for each $\mathbf x,\ell$.

For any joint state $\rho$, let $\sigma_\ell=\langle\ell|\rho|\ell\rangle$ be the positive, unnormalized conditional operator on Bob's space and $\mu_\ell=\operatorname{Tr}\sigma_\ell$. Then $\mu_\ell\geq0$ and $\sum\mu_\ell=1$. For Bob's arbitrary POVM effects $B_{\mathbf b}^{\mathbf y}$, the correlation is \[ p(\mathbf a,\mathbf b\mid\mathbf x,\mathbf y) =\sum_{\ell:\mu_\ell>0}\mu_\ell\, a_{\mathbf a}^{\mathbf x}(\ell) \frac{\operatorname{Tr}(\sigma_\ell B_{\mathbf b}^{\mathbf y})}{\mu_\ell}. \] This is a classical shared-randomness strategy with hidden variable $\ell$ and local response probabilities. Zero-weight terms vanish because a positive operator of trace zero is zero. Therefore any such commuting-Alice construction is bounded by the classical value, whatever state or Bob measurements are chosen. In particular embedding the displayed deterministic Alice table into a commuting quantum measurement and optimizing only Bob cannot yield the desired advantage. A successful quantum route must use noncommuting measurements on both parties, or otherwise escape this explicitly proved restricted class.

The exact finite search in Verification also shows that, with the displayed Alice table fixed, optimizing each of Bob's eight answer triples independently still gives total score $20/64$. This is an actual exhausted restricted search, not a global search over entangled strategies.

Verification and adversarial audit

The script evaluates all 64 input pairs and obtains 20 wins. It enumerates all 27 Bob answers for each of his eight inputs, obtaining best scores $(2,2,2,4,2,2,2,4)$ out of the eight Alice inputs. It also checks all 432 single-question replacements on either party's displayed table, none exceeding 20 wins. These searches do not prove the global classical upper bound or the quantum upper bound. The common-eigenbasis argument proves a separate structural obstruction to one-sided commuting strategies.

Audit corrections.

Kept reported global upper bounds external, distinguished finite table verification from exhaustive global optimization, and excluded independent-product/mixture ansatzes only at their stated scopes.

External computation.

Ran check_feige_three.py under Python 3.14.2 using exact Boolean arithmetic and Fraction: 20/64 wins, fixed-Alice best-response total 20, all 432 one-question alterations checked; hybrid values 9/32 and 729/4096.

Scope and remaining gaps

No entangled strategy with value above $5/16$ or quantum upper certificate at $5/16$ is found. The original threefold question remains unresolved in this attempt; the interval between the cited classical and nonsignalling values remains unclosed.

Integration review.

Codex (GPT-6; exact serving revision not exposed) reviewed the written arguments and expanded layout and source-reference wrappers. Repository metadata and the production web renderer were checked using Python 3.13 and JavaScript with MathJax 3.2.2. This model-assisted review is not independent human proof verification; no PDF compilation or formal proof checking is asserted.

Completion Estimate

COMPLETION: 0% The calculations below the preserved source statement investigate a concrete route, but establish no substantiated new progress toward the entire original target. Known special cases and failed sufficient conditions are not credited as a solution.

References

The original source references and catalogue attribution are preserved in the Sources section above. The following URLs were actually consulted for this attempt; retention of an additional original reference does not claim fresh consultation of it.

• https://ems.press/content/serial-article-files/53598 (consulted 2026-10-11).
• https://arxiv.org/html/2510.08484v3 (consulted 2026-10-11).
• https://arxiv.org/abs/2510.08484 (consulted 2026-10-11).

Model and contributor disclosure

Codex GPT-6 completed written-proof review of this attempt at its declared scope; this is model-assisted review, not independent human proof checking. OpenAI gpt-6-astra, requested reasoning configuration ultra, generated this research attempt on 2026-10-11. The exact serving revision is not exposed. Codex GPT-6 performed orchestration and repository-format review; its exact serving revision is also not exposed. The mathematical route, boundary checks and final adversarial self-audit followed contribution prompts A then B in docs/contribution_prompts.md, after reading CONTRIBUTING.md. Python 3.14.2 assembled the standalone documents; any mathematical computation is specified in Verification. No theorem prover was used. No human mathematical authorship, substantial human proof correction, or independent human proof checking is asserted. Model-assisted review, source retrieval, and structural checks do not establish correctness or novelty. Full private reasoning transcripts are unavailable as reproducibility artifacts; the complete arguments, inputs and outcomes actually used are stated here or in the companion batch audit.

Final status

Unresolved as a new research contribution. This is a concrete route investigation, not a claim of new complete resolution or a current-status certification. A known answer reproduced for context does not become a new contribution. The catalogue's source-status label is preserved; this attempt status does not assert that a source-labelled solved question is open.