{"id":"0b3d8fc2-a093-4c55-8b47-c5fb698c9dca","arxiv_id":"2411.15536","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"The authors identify four-player CHSH games in which a GHZ state yields a 22.5% quantum advantage over classical play, and another game where a W state is the better resource.","lead":"This paper finds two new four-player versions of the classic CHSH game where quantum strategies clearly beat any classical strategy. One game achieves a 22.5% advantage with a GHZ entangled state, more than double the usual 10% gap, and the results were tested on a real quantum computer.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Classical bound for Eq. (9) is invalid: the all-zero deterministic strategy wins 11/16 = 0.6875, so the advertised 22.5% quantum gap is unsupported.","rationale":"The reader's conditional approval missed a checkable error in the classical baseline. The paper's own Section 5 admits the state search was non-exhaustive, but the classical bound is not a matter of sampling: a single valid deterministic strategy already beats the stated maximum. I verified the f table by hand: f = 0 occurs on 11 of the 16 question tuples. Thus the headline gap is at most 16.6 percentage points against the trivial strategy, and the abstract's 22.5% figure is wrong. This does not destroy all value, since a quantum advantage may still exist, but the central claim as stated fails. The reader's weakest_assumption, that SLOCC sampling does not prove optimality, is real but secondary; the classical-bound error is determinative. Hence the verdict should move from CONDITIONAL to REJECT.","tokens_in":36,"tokens_out":9786,"duration_ms":392506,"concrete_test":"Run the exhaustive classical search described in Section 2.2 for Eq. (9) under uniform questions: enumerate all 2^8 deterministic strategies and compute the exact maximum win rate. Separately evaluate the all-zero strategy, which wins on the 11 tuples with f = 0, giving 0.6875. If the true maximum is at least 0.6875, the paper's 0.6225 bound and the 22.5% gap are disproven. This check takes seconds and settles the central numerical claim.","verdict_should_be":"REJECT","load_bearing_attack":"The central claim, a 22.5% quantum advantage for the GHZ game of Eq. (9), rests on the assertion in Section 3.2 that no classical strategy beats 0.6225. This bound is false. For Eq. (9), f(w,x,y,z) = xyz + xyw + xzw + yzw + wxyz (with + as XOR/OR, the value table is the same). Direct evaluation gives f = 0 on 11 of the 16 equally likely question tuples: for w = 0 only (x,y,z) = (1,1,1) gives 1; for w = 1 the 1-valued tuples are exactly 011, 101, 110, 111. The valid deterministic strategy where every player always answers 0 satisfies a xor b xor c xor d = 0 and therefore wins on all 11 tuples where f = 0, i.e., with probability 11/16 = 0.6875. This exceeds the claimed upper bound of 0.6225, so the reported classical maximum is not the maximum. The true gap to the quantum value 0.8535 is at most 0.166, not 0.225 or 0.231, undermining the headline. Because this bound is the basis for the 'optimal resource' conclusion and the abstract's central quantitative promise, the main result as stated is not supportable.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper generalizes the two-player CHSH game to four players by enumerating Boolean functions f and comparing classical and quantum winning probabilities. It reports a four-player game (Eq. (9)) for which a GHZ-based strategy wins with probability about 0.8535, claimed to exceed the best classical strategy by 22.5%, and a second game (Eq. (10)) where a W-state strategy wins with probability about 0.7499 versus 0.6875 classical. It also tests other four-qubit states from the Verstraete classification and implements both games on IBM quantum hardware.","tokens_in":11844,"tokens_out":15207,"duration_ms":128450,"significance":"The paper's methods—exhaustive enumeration over classical strategies, numerical optimization over single-qubit unitaries, and an experimental demonstration—are a useful contribution, and the public availability of the code is commendable. If the headline advantage were correct, the 22.5% gap would be a notable departure from the roughly 10% gap usually reported for CHSH-type games. However, the headline numerical claim is invalid because the classical bound is computed incorrectly; the corrected gap is still substantial (about 16.6%) but smaller. In addition, the 'optimal' status of the GHZ state is not proven by the numerical scan over a few sampled states, so the strongest claims in the abstract and conclusion need to be moderated.","major_comments":[{"comment":"The claimed classical bound 0.6225 for Eq. (9) is incorrect. The deterministic strategy in which all four players always answer 0 gives a⊕b⊕c⊕d = 0 and wins on every question tuple where f(w,x,y,z) = 0. For the function in Eq. (9), f = 0 on 11 of the 16 equally likely tuples (all seven w=0 tuples except 0111, plus the w=1 tuples 1000, 1001, 1010, 1100), so this strategy wins with probability 11/16 = 0.6875. This exceeds the reported 0.6225, so the claimed 22.5% gap is false. The correct gap to the quantum value 0.8535 is at most 0.166, not 0.225. This error propagates to the abstract, Section 3.2, Table 1, Table 3, Table 6, Section 4.1, Figure 9, and the conclusion, and must be corrected.","section":"§3.2, Eq. (9), Table 1"},{"comment":"The claim that the GHZ state is the optimal quantum resource for Eq. (9) is not established. The evidence is a numerical scan over one representative from each of the nine SLOCC families in the Verstraete classification, with only four random parameter choices for the six parametric families, and the unitary optimization is a heuristic numerical search over the three Euler angles per player and question. This does not rule out states outside the sampled points or better local unitaries. The abstract and conclusion should be weakened to state that GHZ outperforms the classical bound and that no state in the sampled families exceeded its score, rather than asserting optimality.","section":"§3.5, Table 6, Abstract"},{"comment":"The reported classical value for Eq. (9) is inconsistent with the exhaustive enumeration described in Section 2.2, where all 2^8 = 256 deterministic local strategies are supposedly generated and evaluated. The all-zero strategy is among those 256 strategies and wins with probability 0.6875, so the claimed optimum 0.6225 indicates a bug in the classical search code or in the transcription of its output. The released code at the GitHub repository should be corrected and all classical bounds re-derived.","section":"§2.2 vs §3.2"}],"minor_comments":[{"comment":"The right-hand side of Eq. (10) is typeset as (abcd)+(abcd)+(abcd)+(abcd), which is a tautological OR of identical terms; please replace it with the intended Boolean expression, as the game is not well defined as printed.","section":"Eq. (10)"},{"comment":"The sentence describing the complement reduction appears to repeat the same equality twice ('the game defined by f = g and the game defined by f = g'); please clarify that f and NOT f (or g and NOT g) are equivalent when a player can flip an answer.","section":"§2.4.1"},{"comment":"The text states that the GHZ state with g = a⊕b⊕c⊕d has a game score of 26.34%, but this row is missing from Table 7; please add it for completeness.","section":"Table 7"},{"comment":"The label 'a classical strategy that achieves an average gain of 62.25%' should be updated to the optimal classical bound (68.75%) if the corrected value is used; otherwise the comparison with the quantum score is misleading.","section":"Figure 9"},{"comment":"The phrase 'The larger number of variables (players) makes it more difficult to find a deterministic strategy' is not supported: the constant-zero strategy is immediately available and wins with probability 11/16.","section":"§3.2"}],"recommendation":"major_revision","confidential_remarks":"The central quantitative claim (22.5% gap) is based on a clear arithmetic error in the classical bound, and the corrected gap (16.6%) is still interesting but less dramatic. The optimality overclaim for the GHZ state would need to be tempered or replaced by a more precise statement. The paper is not acceptable in its current form, but the core idea and the corrected results are likely publishable after the classical bounds are recomputed and the claims are recalibrated."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The stress-test note is correct, and the reader's conditional verdict is too generous. For Eq. (9), if all four players always answer 0, the right-hand side a⊕b⊕c⊕d is 0, and the left-hand side f(w,x,y,z) = xyz⊕xyw⊕xzw⊕yzw⊕wxyz is also 0 on 11 of the 16 question tuples. That deterministic strategy wins with probability 11/16 = 0.6875, which exceeds the paper's claimed classical upper bound of 0.6225. The claimed 22.5% gap (or the 23.1% you get from their own numbers) is therefore not supported. The gap to the stated quantum value 0.8535 is at most 16.6 percentage points, and it could shrink further if an even better classical strategy exists. The sentence in Section 3.2 saying it is impossible to win classically with probability higher than 0.6225 is simply false.\n\nThat said, the paper is not without merit. The systematic reduction from 65,536 Boolean functions to 3,907 and the scan over all of them for g = a⊕b⊕c⊕d is a legitimate computational contribution. The explicit single-qubit rotation strategies in Tables 1 and 2 are concrete and checkable. The W-state game of Eq. (10), where W beats both the classical bound and GHZ, is a genuine new example, and the paper's game-score comparison across the nine SLOCC families is a useful survey. The authors also ship code and data and ran real IBM hardware, which is credit-worthy.\n\nThe soft spots beyond the central classical-bound error: the \"optimal resource\" language is overclaimed. Section 3.5 tests one random instance per parametric SLOCC family, repeated four times, plus numerical gate-angle optimization. That does not establish optimality over the continuous infinite family of four-qubit states, and the conclusion itself admits exhaustive search was impractical. The IBM experiments also lack error bars or a significance statement, and the W-state experiment sits only about 1.4 percentage points above the classical bound, so those numbers need statistical backing.\n\nThis paper deserves a serious referee because the underlying search method and the W-state result are worth reviewing, but not in its current form. The referee should be specifically asked to recompute the classical bounds. I would not cite it in its current state, but I would look again at a corrected version.","headline":"The 22.5% headline is wrong: for Eq. (9) the constant-zero classical strategy wins 11/16 = 0.6875, so the advertised quantum advantage is unsupported even though the paper's computational scan and W-state game retain real value.","tokens_in":127,"tokens_out":5934,"would_cite":false,"duration_ms":110547,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"For the four-player CHSH game defined by Eq. (9), a GHZ-based strategy wins with probability about 0.8535 versus 0.6225 for any classical strategy, a 22.5% gap; a second game favors the W state.","keywords":["four-qubit CHSH games","quantum games","GHZ state","W state","quantum advantage","Boolean functions","entanglement classification","nonlocal games"],"falsifier":"Systematically optimize the win probability for Eq. (9) over the full parameter range of the six parametric families in the nine-family classification (for example, sweeping a and b finely in the Gabcd family). Finding any parameter choice with a win probability above 0.8535 would refute the claimed optimality of the GHZ state; recovering 0.8535 only along the GHZ orbit would support it.","tokens_in":11357,"feed_emoji":"🎲","tokens_out":8009,"duration_ms":62360,"temperature":0.7,"pith_summary":"This paper generalizes the two-player CHSH nonlocal game to four players by enumerating all relevant Boolean winning conditions and numerically searching for the best quantum strategies. It identifies a game, defined by the equation (xyz)+(xyw)+(xzw)+(yzw)+(wxyz) = a⊕b⊕c⊕d, where four players sharing a GHZ state win with probability about 0.8535, compared with 0.6225 for the best classical strategy. That 22.5% gap is more than double the roughly 10% advantage seen in the two- and three-player versions. A second game is found where a W state wins with probability about 0.7499, beating both the classical bound of 0.6875 and the GHZ state's 0.5727. The games were implemented on an online quantum computer, where the quantum strategies surpassed the classical bounds.","feed_headline":"Four-qubit game widens quantum advantage to 22.5%","feed_subtitle":"GHZ and W states each win a new four-player game; real hardware confirms the advantage.","key_machinery":"The search machinery is exhaustive enumeration of the 3,907 essentially different Boolean functions f(w,x,y,z) (down from 65,536 by symmetry and relevance reductions), paired with numerical optimization of every player's local unitary, each parameterized by three rotation angles, for a fixed shared state and fixed answer function g. The resource space is organized by the nine-family SLOCC classification of four-qubit entanglement, which groups all pure states up to local operations; from each family a representative is optimized, with four random parameter choices for the six parametric families. A 'game score' metric — the percentage of candidate functions f for which a state beats the classical win probability by more than 1% — ranks how often each entanglement type yields a quantum advantage.","core_discovery":"The paper's central claim is that four-player CHSH-type games can exhibit quantum advantages substantially larger than the 10% observed in the two- and three-qubit cases, and that the best resource depends on the game. For Eq. (9), the GHZ state yields a win probability of 0.8535 versus 0.6225 classically, and the authors assert that no other four-qubit state they tested — including critical states and representatives of every family in the nine-family entanglement classification — exceeds this score. For Eq. (10), the W state yields 0.7499 versus 0.6875 classically, while the GHZ state reaches only 0.5727 on the same game, making the W state the better resource there. The authors take these results to show that different types of four-qubit entanglement are optimal for different games, with the GHZ state achieving the maximal quantum advantage among all tested resources for games of the form f(w,x,y,z)=a⊕b⊕c⊕d.","pith_inferences":["If the observed growth in the quantum-classical gap continues with player number, analogous searches for n≥5 could reveal even larger advantages, but the paper's numerical method would need a more efficient reduction because the function count grows doubly exponentially.","A certified optimality proof for Eq. (9) would require either an analytical upper bound on the win probability over all four-qubit states or a dense sweep of the parametric families; the present four random samples per family leave open the possibility of a better state just off the sampled grid.","The game-score statistic could be used as a theory-neutral benchmark: entanglement families that score high on many games might be better candidates for device-independent protocols, though the paper does not draw that connection."],"forward_implications":["For Eq. (9), the GHZ strategy wins with probability 0.8535 against 0.6225 classically, so any future four-player CHSH variant with the same g(a,b,c,d)=a⊕b⊕c⊕d and a different f cannot beat this gap unless a state outside the sampled families does so.","For Eq. (10), a W state is superior to a GHZ state, showing that four-qubit entanglement resources are game-specific rather than universally ordered.","The experimental runs on real quantum hardware beat the classical bound for both games, indicating the advantage survives noise at the level of current devices.","The 'game score' tables provide a practical ranking of four-qubit entanglement families by how often they grant a quantum advantage across all games of the studied form."],"supporting_citations":[{"why":"Defines the original CHSH game and its classical and quantum win probabilities, the baseline being generalized.","marker":"[6]"},{"why":"Supplies the three-player CHSH game method and the observation that different entangled states suit different games, extended here to four players.","marker":"[11]"},{"why":"Provides the nine-family SLOCC classification of four-qubit entanglement used to organize the resource space.","marker":"[16]"},{"why":"Supplies the corrected normal forms for those nine families used in the numerical tests.","marker":"[5]"},{"why":"Provides the critical-state candidates (Mermin-Peres, cluster, and L states) tested as alternative resources.","marker":"[14]"},{"why":"Gives the deterministic construction of W states used to prepare the resource on the quantum computer.","marker":"[8]"}],"fun_headline_variants":["Four-player game pushes quantum advantage to 22.5%","GHZ and W states both win four-player quantum games","Quantum advantage jumps to 22.5% in four-qubit game","Four-qubit CHSH games beat 10% advantage on real hardware","New four-qubit games double quantum advantage over classical"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The conclusion that the GHZ state is the best possible resource for Eq. (9) rests on testing one representative from each of the nine entanglement families with only a few random parameter choices per parametric family — a finite numerical scan, not a proof over the continuous space of all four-qubit states.","fun_headline_variants_meta":{"raw":{"variants":["Four-player game pushes quantum advantage to 22.5%","GHZ and W states both win four-player quantum games","Quantum advantage jumps to 22.5% in four-qubit game","Four-qubit CHSH games beat 10% advantage on real hardware","New four-qubit games double quantum advantage over classical"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000186,"raw_usage":{"total_tokens":1296,"prompt_tokens":885,"completion_tokens":411,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":501,"completion_tokens_details":{"reasoning_tokens":324}},"tokens_in":501,"tokens_out":411,"duration_ms":3968,"temperature":1.0,"reasoning_tokens":324,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T14:10:58.711097+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Systematically optimize the win probability for Eq. (9) over the full parameter range of the six parametric families in the nine-family classification (for example, sweeping a and b finely in the Gabcd family). Finding any parameter choice with a win probability above 0.8535 would refute the claimed optimality of the GHZ state; recovering 0.8535 only along the GHZ orbit would support it.","supporting_citations":[{"cited_title":"F Clauser, M","cited_arxiv_id":null,"evidence_quote":"Defines the original CHSH game and its classical and quantum win probabilities, the baseline being generalized."},{"cited_title":"Jaffali and F","cited_arxiv_id":null,"evidence_quote":"Supplies the three-player CHSH game method and the observation that different entangled states suit different games, extended here to four players."},{"cited_title":"Verstraete, J","cited_arxiv_id":null,"evidence_quote":"Provides the nine-family SLOCC classification of four-qubit entanglement used to organize the resource space."},{"cited_title":"Chterental and D","cited_arxiv_id":null,"evidence_quote":"Supplies the corrected normal forms for those nine families used in the numerical tests."},{"cited_title":"Oeding and I","cited_arxiv_id":null,"evidence_quote":"Provides the critical-state candidates (Mermin-Peres, cluster, and L states) tested as alternative resources."},{"cited_title":"Diker,Deterministic construction of arbitraryw states with quadratically increasing number of two- qubit gates, 2022","cited_arxiv_id":null,"evidence_quote":"Gives the deterministic construction of W states used to prepare the resource on the quantum computer."}],"review_version":1}