{"id":"8f8fc67d-f0ab-4374-a02b-52be17c69162","arxiv_id":"2608.05378","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"partial","parameter_count":3,"one_line_summary":"A bipartite nonlocal game is exhibited that has a perfect entangled strategy with a non-maximally entangled state but no perfect maximally entangled strategy, refuting the conjecture that maximally entangled states are complete for pseudo-telepathy.","lead":"Quantum pseudo-telepathy games can sometimes be won perfectly with a non-maximally entangled state even when no maximally entangled state works. This paper exhibits the first such bipartite game, settling a long-standing open question in quantum nonlocality.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The negative half of Theorem 1.1 depends on an unspecified 'slightly strengthened' NPA hierarchy in §4.2; until its added constraints are stated and proven necessary for maximally entangled strategies, the claim cannot be verified from the paper.","rationale":"The constructive half of the paper is sound: Proposition 3.1 gives an explicit state and measurements achieving a perfect entangled strategy for the inner product game, and the unitaries in Appendix A can be checked directly. The load-bearing part is the nonexistence of any perfect maximally entangled strategy, which rests entirely on the tracial NPA hierarchy computation. The reader's weakest-assumption analysis identified exactly this point: the 'slightly strengthened version' of the hierarchy is not specified, so its soundness is not established in the manuscript. I agree with that assessment. A rational certificate and a Lean formalization are genuinely strong forms of evidence, but only if the exact constraints used are available and are proven to be necessary consequences of the maximally entangled strategy condition. Because the paper omits the constraints and does not pin a repository revision, the certificate currently proves infeasibility of an unknown relaxation rather than nonexistence of a perfect maximally entangled strategy. The proposed test—extract the added constraints from the Lean code and prove each is satisfied by maximally entangled strategies—would settle the concern. Until then, conditional acceptance is the appropriate verdict, so no change to the reader's verdict is needed.","tokens_in":7911,"tokens_out":4260,"duration_ms":43861,"concrete_test":"Check out the GitHub repository at the commit used by the author, compile the Lean project, and locate the declaration `noPerfectMaximallyEntangledStrategy` together with the definition of the strengthened hierarchy. Then independently derive the added constraints from the definition of a strategy on |Ω⟩ using Lemma 2.1, and verify in Lean that each added constraint is satisfied by every tracial state induced by such a strategy. If any added constraint is not derivable from the exact maximally entangled strategy conditions, the infeasibility certificate may be ruling out an over-restricted set and the nonexistence proof is unsound.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 4.2 rules out perfect maximally entangled strategies by testing infeasibility of a 'slightly strengthened version' of the n=4 tracial NPA hierarchy. The manuscript never states the added constraints, nor does it prove that every correlation realized by measuring a maximally entangled state satisfies them. A rational infeasibility certificate is only evidence about the strengthened relaxation: if one added constraint is not valid for all maximally entangled strategies, the SDP can be infeasible even though a perfect maximally entangled strategy exists. The claimed Lean formalization would resolve this only if its theorem statement exactly matches the game G_{4,3,6,6} and if the repository is pinned to the verified commit; the paper gives neither the strengthened constraints nor a commit hash. Thus the central negative claim is currently an uncheckable black box, and the positive existence direction of Proposition 3.1 does not compensate for this gap. The missing piece is not the use of computation or AI, but the missing statement and soundness proof of the additional hierarchy constraints.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a new class of nonlocal games called inner product games and exhibits a specific instance G_{4,3,6,6} (4 inputs for Alice, 3 for Bob, 6 outputs each). The positive direction of the main theorem is an explicit perfect entangled strategy using a non-maximally entangled state of local dimension 6, obtained from Proposition 3.1 with S = diag(1,1,2,2,2,2). The negative direction claims that no perfect maximally entangled strategy exists for this game; this is supported by an infeasibility certificate for a 'slightly strengthened version' of the n=4 tracial NPA hierarchy, with the proof allegedly formalized in Lean. If correct, Theorem 1.1 disproves the conjecture that maximally entangled states are complete for bipartite pseudo-telepathic games.","tokens_in":8107,"tokens_out":2621,"duration_ms":25300,"significance":"The result, if fully verified, would settle a longstanding open question in quantum nonlocality and would be a notable contribution. The paper has several strengths: the inner product game construction is elegant, Proposition 3.1 gives a clean analytic proof of the positive existence direction, the game matrices are explicitly listed, and the use of a machine-checked certificate is a valuable reproducibility feature. The main weakness is that the negative direction depends on an unspecified strengthening of the NPA hierarchy in Section 4.2; until the added constraints are stated and proven valid for all maximally entangled strategies, the central nonexistence claim cannot be independently checked from the manuscript alone. This is a load-bearing gap that prevents the paper from being accepted in its current form.","major_comments":[{"comment":"The central negative claim rests on an undefined object: the 'slightly strengthened version' of the n=4 tracial NPA hierarchy. The paper neither states the additional constraints nor proves that every correlation arising from a perfect maximally entangled strategy satisfies them. If even one added constraint is not valid for all maximally entangled strategies, the SDP infeasibility certificate would not rule out the existence of such a strategy. Please include the exact hierarchy constraints, or a precise reference to where they are defined, together with a soundness proof that they hold for any perfect maximally entangled strategy of G_{4,3,6,6}.","section":"§4.2"},{"comment":"The claimed Lean formalization does not resolve the gap as presented. The theorem statement is given only as 'noPerfectMaximallyEntangledStrategy (d : Nat)' with no formal statement in the paper, and the GitHub repository is not pinned to a commit. A reader cannot determine whether the verified theorem exactly matches the game G_{4,3,6,6} and the strengthened hierarchy. Please provide, in the manuscript or supplementary material, the precise formal statement, the version/commit of the repository, and an explanation of how the formalized statement implies the nonexistence of a perfect maximally entangled strategy for the game.","section":"§4.2, Theorem 1.1"},{"comment":"The soundness of the tracial NPA hierarchy itself is only invoked, not stated. For the unstrengthened hierarchy, the implication 'infeasibility at level n implies no tracial state' is standard, but the paper's conclusion requires that the strengthened constraints are necessary conditions on the tracial state derived from a maximally entangled strategy. Please make this chain of implications explicit and identify exactly where the added constraints enter the formalized proof.","section":"§4.2"}],"minor_comments":[{"comment":"In the displayed calculation, the projector for Bob's measurement appears as 'VY |b⟩ ⟨b|VT y' and later 'V† y'; this is presumably meant to be a consistent notation such as V_y^T |b⟩⟨b| V_y^*. Please correct the typo and align the notation with the text description of Bob applying V_y^T.","section":"§3, proof of Proposition 3.1"},{"comment":"The matrices are presented as raw arrays; it would help readers to state explicitly that the columns are orthonormal and that the displayed entries are real. A short verification script or a note that these are unitary would improve reproducibility.","section":"Appendix A"},{"comment":"The phrase 'slightly strengthened version' is too vague for a result of this importance. Even if the constraints are given elsewhere, please reserve a paragraph to motivate which constraints are added and why they are natural for maximally entangled strategies, analogous to the discussion in [Rus23].","section":"§4.2"},{"comment":"The reference to [Ren+26] is followed by a sentence saying this 'does not address the problem of whether they are always sufficient'; consider clarifying that the new result goes beyond [Ren+26] by showing non-maximally entangled states can be necessary, not merely sufficient.","section":"§1"}],"recommendation":"major_revision","confidential_remarks":"The central claim is potentially important and the positive direction is solid, but the negative direction is currently unverifiable from the manuscript because the strengthened hierarchy constraints are not stated. This is a fixable issue if the author can provide the explicit constraints, their soundness proof, and a pinned Lean repository with a matching theorem statement. If those cannot be supplied, the paper would need to be rejected because the main theorem would rest on an uncheckable computation. I also note that the heuristic search in [Lal25] is cited appropriately and does not itself create circularity."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper does something genuinely new: it gives an explicit bipartite game G_{4,3,6,6} and claims it has a perfect entangled strategy with a non-maximally entangled state but none with a maximally entangled state, disproving a conjecture that has been open for years. The positive half is solid. Proposition 3.1 gives a clean general construction for inner product games, and the specific perfect strategy using the state (2) is explicitly derived. The game itself is fully specified, and the classical infeasibility is checked by exhaustive search. The author is also refreshingly honest about the computational and manufactured nature of the example, including a clear AI-use statement.\n\nThe soft spot is exactly where the stress-test note lands. Section 4.2 rules out perfect maximally entangled strategies by testing a \"slightly strengthened version\" of the n=4 tracial NPA hierarchy, but the added constraints are never stated, and no proof is given that every correlation arising from a maximally entangled strategy satisfies them. Without that, the infeasibility certificate is only evidence about a relaxation that might be too strong. A perfect maximally entangled strategy could still exist even if this particular SDP is infeasible. The claim that the proof is formalized in Lean would close the gap if the theorem statement matches the game and the repository is pinned to a specific commit, but neither is provided in the text. So as written, the central negative claim is unverifiable from the paper itself.\n\nThis gap is serious but fixable. The author needs to spell out the additional hierarchy constraints, prove their validity for maximally entangled strategies, and give a precise reference to the verified code. The paper already contains the right structure; it just omits the load-bearing details.\n\nFor whom is this paper? Anyone working on nonlocality, pseudo-telepathy, or entangled chromatic numbers will want to engage with it. If the negative half is made rigorous, it is a major result. Even with the gap, the inner product game framework and the explicit example are worth taking seriously.\n\nRecommendation: send it to peer review. A competent referee can push the author to supply the missing statements and verification. This is not a desk reject; it is a paper that deserves a serious referee, with the expectation of heavy revision before it is publishable.","headline":"A potentially important counterexample to the completeness conjecture, with a clean positive construction but a load-bearing gap in the negative half that must be fixed before the result is fully verifiable.","tokens_in":8629,"tokens_out":1801,"would_cite":false,"duration_ms":16961,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["81P40","81P45"],"pacs":[],"model":"deepseek-v4-flash","headline":"One quantum game cannot be won with any maximally entangled state","keywords":["pseudo-telepathy","maximally entangled states","inner product games","tracial NPA hierarchy","nonlocal games","semidefinite programming","quantum nonlocality"],"falsifier":"Try to find a perfect maximally entangled strategy for $G_{4,3,6,6}$: because there are only 12 input pairs and 36 output pairs, an exhaustive search over six-dimensional measurements is a concrete computational check. Finding one would contradict Theorem 1.1, and running the unstrengthened level-4 hierarchy and finding it feasible would show the added constraints were doing the work.","tokens_in":7650,"feed_emoji":"🎯","tokens_out":10383,"duration_ms":88401,"temperature":0.7,"pith_summary":"Quantum pseudo-telepathy is the phenomenon in which two separated players can win a nonlocal game with certainty using shared entanglement, even though no classical strategy achieves this. A longstanding question asked whether the shared state can always be taken to be maximally entangled. This paper answers the question negatively, exhibiting a specific bipartite game $G_{4,3,6,6}$ with 4 and 3 inputs and 6 outputs per player. The game has a perfect entangled strategy using a six-dimensional non-maximally entangled state, but the paper proves that no maximally entangled state can win it perfectly. The proof combines a new class of games, called inner product games, with a numerical semidefinite-programming certificate verified in a proof assistant.","feed_headline":"No maximally entangled state can win this quantum game","feed_subtitle":"A 4-by-3-input game settles a long-standing completeness question for pseudo-telepathy.","key_machinery":"The paper's central object is a new class of games called inner product games. For unitaries $U_x$, $V_y$ and a positive semidefinite matrix $S$, players receive $x,y$ and output $a,b$, winning exactly when $\\langle a|U_x^\\dagger S V_y|b\\rangle \\neq 0$; the matrix $S$ is promoted into a non-maximally entangled state $|\\psi\\rangle = (S\\otimes I)|\\Omega\\rangle$ on which a perfect strategy is built. To rule out maximally entangled strategies, the paper uses the tracial NPA hierarchy: a perfect maximally entangled strategy would force the existence of a tracial state on a certain $*$-algebra satisfying constraints for losing outputs, and a slightly strengthened level-4 semidefinite relaxation of this condition is proved infeasible. The infeasibility certificate is a rational 5.6-megabyte file whose validity is formalized in a proof assistant.","core_discovery":"On the paper's own terms, the discovery is Theorem 1.1: there is a bipartite nonlocal game $G_{4,3,6,6}$ with input sets of sizes 4 and 3 and output sets both of size 6 that admits a perfect entangled strategy, but no perfect entangled strategy whose shared state is maximally entangled. The winning strategy uses the state $|\\Phi\\rangle = \\frac{1}{\\sqrt{18}}(|1\\rangle|1\\rangle+|2\\rangle|2\\rangle+2|3\\rangle|3\\rangle+2|4\\rangle|4\\rangle+2|5\\rangle|5\\rangle+2|6\\rangle|6\\rangle)$. The game is pseudo-telepathic: it has no perfect classical strategy either, as verified by exhaustive search. This gives the first counterexample to the conjecture that maximally entangled states are complete for bipartite pseudo-telepathy.","pith_inferences":["An inference beyond the paper: the numerics suggesting self-testing point toward a stronger statement, that any perfect entangled strategy for $G_{4,3,6,6}$ must essentially use the non-maximally entangled state (2), not merely some non-maximal state.","Another extension: because attempts at dimensions 3, 4, and 5 failed before dimension 6 succeeded, there may be a minimal local dimension at which a bipartite pseudo-telepathic game forces non-maximal entanglement; locating that threshold is a natural question.","If the reverse implication to the entangled chromatic number can be established, this game is a candidate to produce the first separation between the entangled and quantum chromatic numbers of a graph, connecting the result to zero-error information theory.","A computational extension: the same tracial-hierarchy pipeline could be modified to prove stronger quantitative statements, such as lower bounds on the minimum Schmidt coefficient needed for any perfect strategy."],"forward_implications":["The conjecture that maximally entangled states are complete for bipartite pseudo-telepathy is false; the counterexample already appears at local dimension 6 with 4 and 3 inputs and 6 outputs per player.","The class of inner product games carries a universal perfect-strategy construction: for any such game, the state $(S\\otimes I)|\\Omega\\rangle$ wins perfectly, so the class is a general source of pseudo-telepathic games.","Because $G_{4,3,6,6}$ has no perfect classical strategy, it is a genuine pseudo-telepathy witness in which non-maximal entanglement is essential for a perfect quantum win.","The proof supplies a rational, machine-checkable infeasibility certificate for the level-4 tracial NPA relaxation, demonstrating that nonexistence of maximally entangled strategies can be certified rigorously even when the underlying game is found numerically."],"supporting_citations":[{"why":"Formulates the open problem of whether maximally entangled states are always sufficient for pseudo-telepathy, which this paper answers in the negative.","marker":"[Man15]"},{"why":"Shows how to impose maximally-entangled-state constraints on the NPA hierarchy, the template for the proof's tracial variant.","marker":"[Rus23]"},{"why":"Provides the sum-of-squares hierarchy on which the tracial relaxation is built.","marker":"[NPA08; Doh+08]"},{"why":"Shows the general existence problem for the relevant tracial states is undecidable, motivating the finite-level relaxation.","marker":"[Slo19]"},{"why":"Supplies the heuristic enumeration and pruning procedure used to find the game $G_{4,3,6,6}$.","marker":"[Lal25]"},{"why":"The proof assistant in which the nonexistence certificate was formalized.","marker":"[MU21]"}],"fun_headline_variants":["Quantum game disproves maximal entanglement completeness","Pseudo-telepathy game without maximally entangled state","Max entanglement not enough for some quantum games","Quantum game shows max entanglement isn't necessary","Entanglement limits in pseudo-telepathy exposed"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof rests on the assumption that every extra restriction added to the level-4 numerical search is genuinely satisfied by any perfect strategy that uses a maximally entangled state; if one restriction is too strong, the infeasibility certificate would not show what the theorem claims.","fun_headline_variants_meta":{"raw":{"variants":["Quantum game disproves maximal entanglement completeness","Pseudo-telepathy game without maximally entangled state","Max entanglement not enough for some quantum games","Quantum game shows max entanglement isn't necessary","Entanglement limits in pseudo-telepathy exposed"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000468,"raw_usage":{"total_tokens":2263,"prompt_tokens":809,"completion_tokens":1454,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":425,"completion_tokens_details":{"reasoning_tokens":1385}},"tokens_in":425,"tokens_out":1454,"duration_ms":10433,"temperature":1.0,"reasoning_tokens":1385,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T14:16:44.921590+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Try to find a perfect maximally entangled strategy for $G_{4,3,6,6}$: because there are only 12 input pairs and 36 output pairs, an exhaustive search over six-dimensional measurements is a concrete computational check. Finding one would contradict Theorem 1.1, and running the unstrengthened level-4 hierarchy and finding it feasible would show the added constraints were doing the work.","supporting_citations":[],"review_version":1}