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REVIEW 3 major objections 5 minor 16 references

Four-Qubit CHSH Games

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2411.15536 v1 pith:Z4GELFTB submitted 2024-11-23 quant-ph math-phmath.MP

Joaquim Jusseau , Hamza Jaffali , Frédéric Holweck This is my paper · ORCID
classification quant-phmath-phmath.MP
keywords four-qubitCHSHgamesquantumGHZstateWadvantageBooleanfunctionsentanglementclassificationnonlocal
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

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.

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 (3)
  1. [§3.2, Eq. (9), Table 1] 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.
  2. [§3.5, Table 6, Abstract] 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.
  3. [§2.2 vs §3.2] 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.
minor comments (5)
  1. [Eq. (10)] 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.
  2. [§2.4.1] 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.
  3. [Table 7] 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.
  4. [Figure 9] 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.
  5. [§3.2] 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.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: win probabilities are computed by direct exhaustive search and numerical optimization, not constructed from the conclusions; only a minor non-load-bearing self-citation and a separate correctness concern about the classical bound.

full rationale

The derivation chain is self-contained. For a fixed four-qubit state and a fixed winning condition of the form f(w,x,y,z)=g(a,b,c,d), the classical optimum is obtained by enumerating all deterministic local strategies (Section 2.2), and the quantum optimum is obtained by numerically optimizing the single-qubit Euler angles of Eq. (7) and evaluating the Born-rule probability of the measurement outcomes. The games of Eqs. (9) and (10) are selected from the reduced list of 3,907 Boolean functions after the symmetry reductions of Section 2.4; the selection does not set the reported probabilities by construction. The strategies in Tables 1 and 2 are explicit unitaries, and the quoted success probabilities are computed from those unitaries and the chosen GHZ or W state. The claim that GHZ is optimal for Eq. (9) rests on a numerical scan over representative states from the nine SLOCC families (Section 3.5) rather than a proof; that is a limitation of evidence strength, not a circular identification. The paper does cite the authors' own prior work [11] for the search methodology, the unitary parametrization, and the three-qubit CHSH games, but none of the central four-qubit numbers is imported from that citation, so the self-citation is minor and not load-bearing. Separately, there is a correctness concern that should be weighed independently of circularity: the asserted classical bound 0.6225 for Eq. (9) appears inconsistent with the all-zero deterministic strategy, which satisfies a xor b xor c xor d = 0 and therefore wins on the 11 of 16 inputs where the left-hand side of Eq. (9) is 0, i.e. with probability 0.6875. If that is correct, the headline quantum gap needs to be recomputed against the true classical optimum. This is a correctness issue, not a circularity of the derivation.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central claims rest on standard nonlocal-game modeling and the SLOCC classification, plus numerically optimized strategy angles. No new physical entities are introduced. The main burden is that 'optimality' of the GHZ state for Eq. (9) is inferred from a finite scan, not proven.

free parameters (3)
  • Gate angles (theta, phi, lambda) per player per question = Tables 1 and 2, 24 angles per game
    Numerically optimized to maximize the win probability for a given game and state; the reported quantum gains are the optimized values.
  • Random SLOCC family parameters = unspecified, repeated four times
    In Section 3.5, parameters a,b,c,d in the parametric four-qubit families are chosen randomly to sample the families; the resulting gains depend on these choices.
  • Game score threshold = 1%
    The 'game score' definition in Section 3.6 counts functions with quantum gain higher than 1% relative to classical gain; this threshold is chosen by the authors.
assumptions (4)
  • domain assumption The optimal quantum strategy for a game can be found by optimizing the three-angle parameterization of single-qubit unitaries in Eq. (7) followed by measurement in the computational basis.
    Invoked in Section 2.3; standard for nonlocal games, but the paper does not prove that strategies outside this family (e.g., adaptive measurements) could not do better.
  • domain assumption Classical strategies can be restricted to deterministic functions hi(xi) without loss of optimality.
    Used in Section 2.2 to bound the classical win probability by exhaustive search over 2^(2n) strategies; randomized strategies are convex mixtures of deterministic ones.
  • standard math Equivalence reductions in Section 2.4 (complementing f or flipping question bits) preserve the optimal classical and quantum gains.
    Section 2.4.1; flipping a player's question bit and adjusting the answer maps winning strategies bijectively.
  • standard math Every four-qubit pure state belongs to one of the nine SLOCC families of Verstraete et al. [16].
    Used in Section 3.5 to select representative states; this is an established classification result cited from [16].

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Cite this review

Pith. "Pith review of Four-Qubit CHSH Games." pith.science (2026). https://pith.science/paper/Z4GELFTB

@misc{pith2026241115536,
  author       = {Pith},
  title        = {Pith review of: Four-Qubit CHSH Games},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Z4GELFTB}},
  note         = {Machine review of arXiv:2411.15536}
}
abstract

In this paper, the CHSH quantum game is extended to four players. This is achieved by exploring all possible 4-variable Boolean functions to identify those that yield a game scenario with a quantum advantage using a specific entangled state. Notably, two new four-player quantum games are presented. In one game, the optimal quantum strategy is achieved when players share a $GHZ$-state, breaking the traditional 10\% gain observed in 2 and 3 qubit CHSH games and achieving a 22.5\% gap. In the other game, players gain a greater advantage using a $W$-state as their quantum resource. Quantum games with other four-qubit entangled states are also explored. To demonstrate the results, these game scenarios are implemented on an online quantum computer, and the advantage of the respective quantum resource for each game is experimentally verified.

Figures

Figures reproduced from arXiv: 2411.15536 by the authors.

Figure 1
Figure 1. Overview of the application of an n-player quantum strategy. (7) Ui,xi (θi,xi , ϕi,xi , λi,xi ) =   cos  θi,xi 2  −e jλi,xi sin  θi,xi 2  e jϕi,xi sin  θi,xi 2  e j(ϕi,xi+λi,xi ) cos  θi,xi 2    with j the complex imaginary unit. 2.4. Problem’s reduction. Recall that there are 2 16 = 65, 536 possible functions from B 4 to B. Therefore, an exhaustive analysis would require testing 65, 5362 equations in or… view at source ↗
Figure 2
Figure 2. CHSH game setup with 4 players: The referee sends a question w to J1, x to J2, y to J3 and z to J4. The players win the game if and only if their answers (a, b, c, d) satisfy the binary equation f(w, x, y, z) = g(a, b, c, d) that defines the game. The referee sends respectively the binary questions w, x, y, and z, to J1, J2, J3 and J4, they apply their strategy, and they answer respectively a, b, c and d (see [PITH… view at source ↗
Figure 3
Figure 3. Evolution of the quantum gain of Labc2 with c = 0 for Eq. (9) with a and b between −9 and 9. 2D contour plot of the Quantum Score of Labc2 with c = 0 for Eq. (9) with a and b between −9 and 9 [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: Evolution of the quantum gain of La2b2 for Eq. (9) with a and b between −9 and 9. 2D contour plot of the Quantum Score of La2b2 on the Eq. (9) with a and b between -9 and 9. In the case of La203 L 1 = a(|0000⟩ + |1111⟩) + (|0011⟩ + |0101⟩ + |0110⟩) the |GHZ⟩ state is a…
Figure 5
Figure 5. Figure 5: Evolution of the Quantum of the Gain of La203 L 1 with a increasing from 2 to 10 approaches |GHZ⟩ state. 3.7.2. Retrieving |MP⟩ Gain. Let us now consider the parametric states Lab3 and La4 when used with Eq. (9). We first observe that |MP⟩ belongs to the Lab3 family fo…
Figure 6
Figure 6. Figure 6: Evolution of the quantum gain of Lab3 for Eq. (9) with a and b between −9 and 9. 2D contour plot of the Quantum Score of Lab3 for Eq. (9) with a and b between -9 and 9 [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]
Figure 7
Figure 7. Figure 7: illustrates this fact as one sees the gain to converge to 0.6767 when a increases. lima→∞ Gain(La4 ) = Gain(|MP⟩) = 0.67677 [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]
Figure 8
Figure 8. Figure 8: Quantum circuit used for applying the quantum strategy with |GHZ⟩. The optimal angles of the local unitary transformations depend on the question sent to the players and are reported in [PITH_FULL_IMAGE:figures/full_fig_p014_8.png]
Figure 9
Figure 9. Figure 9: Histogram of the percentage of victory as a function of the question sent to the four players in the game defined by Eq. (9). In green a classical strategy that achieves an average gain of 62.25% of victory. In blue, the quantum gain with an average score of 82.19%, th…
Figure 10
Figure 10. Figure 10: Quantum circuit used for applying the quantum strategy with |W⟩. The opti￾mal angles of the local unitary transformations depend on the question sent to the players and are reported in [PITH_FULL_IMAGE:figures/full_fig_p015_10.png]
Figure 11
Figure 11. Figure 11: Histogram of the percentage of victory as a function of the question sent to the four players for the game defined by Eq. (10). In green a classical strategy that achieves an average gain of 68.75% of victory. In blue, the quantum gain with an average score of 70.13%,…

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