REVIEW 3 major objections 4 minor 25 references
Maximally entangled states are not complete for pseudo-telepathy
T0 review · 3 major / 4 minor · reviewed 2026-08-08 · deepseek-v4-flash
Pith's one-line read One quantum game cannot be won with any maximally entangled state
desk verdict 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. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
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.
What would settle it
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.
Extended reading notes
Core claim
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.
Load-bearing premise
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.
Editorial extensions
If this is right
- 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.
Reading between the lines
- 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.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
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.
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 (3)
- [§4.2] 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}.
- [§4.2, Theorem 1.1] 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.
- [§4.2] 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.
minor comments (4)
- [§3, proof of Proposition 3.1] 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.
- [Appendix A] 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.
- [§4.2] 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].
- [§1] 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.
Circularity Check
No significant circularity: the game is explicitly constructed, the perfect strategy is derived self-contained, and the negative claim rests on an external machine-checked certificate; the unspecified 'slightly strengthened' hierarchy is a verification gap, not a circular reduction.
full rationale
The derivation chain is not circular. The game G_{4,3,6,6} is explicitly specified by S=diag(1,1,2,2,2,2) and the unitaries in Appendix A; Proposition 3.1 constructs a perfect strategy with state |ψ>=(S⊗I)|Ω>, and the win-probability identity P(a,b|x,y)=|⟨a|U_x† S V_y|b⟩|² is computed directly from the definition of the game. The nonexistence of a perfect maximally entangled strategy is not obtained by fitting a parameter to that conclusion: it is delegated to a rational infeasibility certificate in a 'slightly strengthened' n=4 tracial NPA level, which the paper says was formalized in Lean. The added constraints are not stated in Section 4.2, so the soundness of that strengthening is an unverified gap: if a constraint were too strong, the SDP infeasibility would not imply nonexistence of a perfect maximally entangled strategy. That is a correctness/verifiability concern, not circularity, because the paper does not define the target state in terms of the certificate or rename a fitted quantity as a prediction. The only self-citation, [Lal25], is invoked as an analogy for the heuristic search that produced the example and is not load-bearing for the theorem. The theorem therefore does not reduce to its own inputs by construction.
Assumptions & free parameters
free parameters (3)
- S matrix =
diag(1,1,2,2,2,2)
- Alice unitaries U1..U4 =
explicit matrices in Appendix A
- Bob unitaries V1..V3 =
explicit matrices in Appendix A
assumptions (5)
- standard math Schmidt decomposition theorem: any pure bipartite state can be written with diagonal Schmidt coefficients.
- standard math Lemma 2.1 identities: (M⊗I)|Ω>=(I⊗M^T)|Ω> and <Ω|M⊗N|Ω>=tr(M N^T).
- domain assumption Soundness of the tracial NPA hierarchy: if no pseudo-state exists up to word length n, then no tracial state satisfying the same constraints exists.
- ad hoc to paper Validity of the 'slightly strengthened' constraints added to the n=4 hierarchy for all perfect maximally entangled strategies.
- domain assumption The Lean theorem noPerfectMaximallyEntangledStrategy faithfully encodes the SDP infeasibility proof.
Cite this review
Pith. "Pith review of Maximally entangled states are not complete for pseudo-telepathy." pith.science (2026). https://pith.science/paper/A7TE5H64
@misc{pith2026260805378,
author = {Pith},
title = {Pith review of: Maximally entangled states are not complete for pseudo-telepathy},
year = {2026},
howpublished = {\url{https://pith.science/paper/A7TE5H64}},
note = {Machine review of arXiv:2608.05378}
}
read the original abstract
One of the longstanding open problems in quantum nonlocality is to determine if maximally entangled states are complete for bipartite pseudo-telepathic games: namely, if every nonlocal game which admits a perfect entangled strategy admits such a strategy which uses a maximally entangled state. We exhibit a counterexample to this in the form of a bipartite nonlocal game with input sets of size 4 and 3 and output sets both of size 6. This game is part of a new class of nonlocal games, which we call inner product games, which could be of independent interest.
Reference graph
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Reviewed August 8, 2026 · model on record in the stance chip above.
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