{"id":"8b1760b4-986f-4252-8e9d-7f3ca09cb92a","arxiv_id":"2505.05994","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A synchronous game that robustly certifies a strategy under maximally entangled projective assumptions robustly certifies the same strategy for all finite-dimensional strategies, with polynomially related robustness.","lead":"This paper proves that robust self-tests for quantum games which assume maximally entangled states automatically work for all strategies, without that physical assumption. The authors apply this to the Quantum Low Degree Test and obtain an efficient test for n-qubit states and Pauli operators.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proofs of Theorem 4.2 and Lemma 5.2 rely on min(Λ) being attained via a false 'Λ is closed' claim; spectral projections are only right-continuous, so Λ can be of the form (a,b] and the construction needs a patch.","rationale":"The reader's weakest assumption was the finite-dimensionality restriction, which is explicitly stated and is a scope limitation rather than an internal gap. My stress-test identifies a different, more proof-specific issue: two central arguments select a minimum of a set Λ of spectral parameters, and the stated justification that Λ is closed is false because spectral projections χ_{≥λ}(ρ_A) are only right-continuous and the normalized winning probability can drop discontinuously at an eigenvalue of ρ_A. This affects both the decomposition into orthogonal ME strategies in Theorem 4.2 and the spectral-gap lower bound in Lemma 5.2, which is then used in Theorems 5.4 and Corollary 5.5. Since the paper has no machine-checked verification, this unproven selection step is a real correctness risk. However, the issue appears repairable: one can choose any point near the infimum of the good interval and absorb the small measure of omitted λ's into the existing error terms, so I would not reject the paper outright. The verdict should be CONDITIONAL — accept once the min(Λ) step is either justified or replaced by an approximate-minimum argument that preserves the stated bounds.","tokens_in":43272,"tokens_out":25403,"duration_ms":277938,"concrete_test":"Test the 'Λ is closed' assertion on the explicit example H_A = C^2, ρ_A = diag(0.99, 0.01), with a 2-outcome PVM whose first basis vector gives winning probability 1 and the second gives 0; compute Λ = {λ : ω(S'_λ) ≥ 1 − √α − ε} and verify it is (0.01, 0.99], so no minimum exists. Then check whether the two uses of min(Λ) in Theorem 4.2 and Lemma 5.2 go through if min is replaced by inf and an arbitrarily small μ-measure of uncovered λ's is added to the error bounds; if yes, the central claim survives with a revised proof, if no the result is unproved.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central lifting argument has a concrete fixable gap in the spectral decomposition step. In Theorem 4.2, Claim 4.3 defines Λ via inequalities involving ω(S'_λ), then says Λ is closed and sets λ1 = min(Λ); Lemma 5.2 similarly sets λ0 = min(Λ), 'which exists because Λ is closed.' But P_λ = χ_{≥λ}(ρ_A) is only right-continuous, not continuous, and the normalized winning probability ω(S'_λ) can jump downward when λ crosses an eigenvalue of ρ_A, since P_λ then includes an additional, possibly bad, eigenspace. In a finite example with ρ_A = diag(0.99, 0.01), where the rank-one λ > 0.01 subspace supports a perfect PME strategy and the full λ ≤ 0.01 space wins with probability 0, the good set is Λ = (0.01, 0.99] with μ(Λ) = 0.98; it has no minimum. Thus the claimed existence of λ1/λ0 is not justified as written. The dyadic partition in Theorem 4.2 and the density-closeness bound in Lemma 5.2 — which feeds Theorem 5.4 and hence Corollary 5.5 — depend on this choice. The gap appears repairable by taking an infimum or any point near it and absorbing the omitted μ-measure into the error terms, but as written the construction is not well-defined.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves a lifting theorem for robust self-testing of synchronous games: if a synchronous game robustly self-tests a perfect PME strategy within the PME class, then it robustly self-tests that strategy within all finite-dimensional POVM strategies, with polynomially related robustness. The proof proceeds in three stages: (i) translating ME/PME strategies into tracial von Neumann algebras and relating the two notions of local dilation; (ii) decomposing an approximately perfect strategy into orthogonal ME strategies via Vidick's theorem and using PME robust self-testing plus a spectral-gap assumption on the game polynomial to construct a dilation; and (iii) showing that the spectral gap is automatically polynomially lower-bounded under PME robustness. The Quantum Low Degree Test is shown to have spectral gap d/2, and the paper applies the lifting result to obtain an efficient n-qubit test. The paper explicitly restricts to finite-dimensional strategies in Section 2.1.","tokens_in":43542,"tokens_out":19521,"duration_ms":195056,"significance":"If the main theorem is correct, it substantially broadens the applicability of PME-only robust self-testing results: every PME-level robust self-test for a synchronous game becomes a full robust self-test for POVM strategies with only polynomial loss. This is directly relevant to the program of upgrading MIP*=RE-style soundness proofs from PME strategies to physically meaningful general strategies, and the QLDT application gives a concrete efficient qubit test. The paper contains several strengths: explicit constants in Lemmas 2.8 and 3.2, a detailed appendix proving Lemma 3.2, and an exact spectral-gap computation for the Quantum Low Degree Test in Theorem 6.1. The main caveat is the spectral-decomposition gap described below, which is localized and appears repairable by a standard limiting argument.","major_comments":[{"comment":"The set Λ defined in Claim 4.3 need not be closed, and it may have no minimum. The spectral projections P_λ = χ_{≥λ}(ρ_A) are only left-continuous in λ; when λ crosses an eigenvalue of ρ_A, P_λ jumps and the winning probability ω(S_λ) can drop, so Λ can be of the form (a,b]. The recursive definition λ1 = min(Λ), λ_{i+1} = min(Λ \\ ⋃Λ_j) is therefore not well-defined as written. This is not merely cosmetic: the estimates in Claims 4.4 and 4.5 use that each λ_i lies in Λ and that the λ_i have the stated dimension-halving property. The gap appears repairable by taking λ_i sufficiently close to the relevant infimum and absorbing the μ-measure of the omitted boundary into the error terms, noting that dμ(λ) = Tr(P_λ)dλ is absolutely continuous with respect to Lebesgue measure; however, this patch must be written out. As it stands, the proof of Theorem 4.2, and hence Theorem 4.6, is incomplete.","section":"Theorem 4.2 and Claim 4.3"},{"comment":"The same issue recurs in Lemma 5.2. After defining Λ = {λ ≥ 0 | ω(S'_λ) ≥ 1 − √α − ε}, the proof states \"Let λ0 = min(Λ), which exists because Λ is closed.\" This is false for the same reason: S'_λ is built from P_λ, and ω(S'_λ) can jump when λ passes an eigenvalue of ρ_A, so Λ can be open on the left and have no minimum. The subsequent estimate ∥ρ_A − ρ_{λ0}∥₁ ≤ 4κ(√α + ε)² + √α, which yields Equation (5.1), depends on this choice. Since Lemma 5.2 is used in Theorem 5.4 to lower-bound the spectral gap, this is a load-bearing point. The fix is the same as in the previous comment: choose λ0 in Λ sufficiently close to inf Λ, or use a limiting argument, and add the corresponding μ-measure error to the bound.","section":"Lemma 5.2"}],"minor_comments":[{"comment":"In the paragraph introducing the strategies S_λ, the text reads S_λ = (|ψ_λ⟩, P_λ A P_λ), but it should be P_λ A' P_λ where A' is the projective strategy from Lemma 2.9; the subsequent claims use A' throughout. Please correct this notation.","section":"Theorem 4.2 setup"},{"comment":"The abstract and introduction say that the result holds for \"all strategies\", while Section 2.1 states \"we assume all strategies employ finite-dimensional systems.\" Please qualify the abstract to say \"all finite-dimensional POVM strategies\" so that the scope is not overstated.","section":"Abstract and Section 2.1"},{"comment":"Lemma 2.9 uses several O(·) bounds without explicit constants. Since the paper elsewhere takes care to provide explicit constants, it would improve reproducibility to state that the constants in Lemma 2.9 are universal and fixed, or to give the explicit polynomial expressions.","section":"Lemma 2.9"},{"comment":"In the proof of the numerical claim near Equation (5.3), the sentence \"where the final step involves some manipulations of fractions\" is vague; please expand the fraction manipulation so that the bound 1/16 + 12√2 can be verified directly.","section":"Theorem 5.4 proof"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the scope of the journal and addresses an important question. The central claim is defensible, and the main technical gap identified above is localized and appears repairable; I do not see a basis for rejection. I would encourage the authors to implement the spectral-minimum patch explicitly and to check whether the same issue appears anywhere else in the proof. Once the patch is in place, the paper should be suitable for publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's the one-line take: this is a genuine quantitative upgrade over the qualitative lifting results in [Zha24, Kar25], and the QLDT spectral gap computation is a clean payoff. But there is a real, localized gap in the central proof that needs patching before the main theorem is fully rigorous.\n\nWhat's new and good: Theorem 1.2/Corollary 5.5 gives a polynomial, game-size-independent conversion from PME-robust self-tests to general finite-dimensional POVM self-tests. That is exactly what the MIP*=RE toolkit wants. Lemma 3.2's equivalence between Hilbert-space and von Neumann algebraic dilation distances is a substantial technical step, with explicit constants and an appendix proof. The proof structure is laid out with claims and explicit error terms, and Section 6's exact spectral gap d/2 for the Quantum Low Degree Test is a nice result. The paper is also honest about working in finite dimension and about importing Vidick's decomposition and de la Salle's orthogonalization.\n\nThe soft spot: the stress-test note is correct. In Theorem 4.2, Claim 4.3 defines Λ by inequalities on ω(S_λ) and a commutator expectation, where S_λ uses P_λ = χ_{≥λ}(ρ_A). The proof then takes λ_1 = min(Λ); Lemma 5.2 does the same with λ_0 = min(Λ), explicitly citing closedness of Λ. But P_λ is right-continuous, not continuous, and λ ↦ ω(S_λ) can jump downward when λ crosses an eigenvalue of ρ_A. The good set can be an interval open at the lower end, e.g. (0.01, 0.99] in a two-eigenvalue example, so min(Λ) need not exist. This is a real gap in the construction of the orthogonal decomposition and in the density-closeness argument that feeds Theorem 5.4 and Corollary 5.5. It looks repairable—take an infimum, or a point approaching it from above, and absorb the omitted μ-measure into the errors—but as written the proof is not well-defined. A referee should ask for this patch, and the authors should also spell out the exponents in the 'poly' bounds in Lemma 2.9.\n\nWho it's for: specialists in self-testing, device-independent certification, and quantum complexity. A serious referee should engage; this deserves review, not desk rejection, and likely acceptance after the gap is fixed.","headline":"Strong quantitative lifting theorem with a real but likely patchable gap in the spectral decomposition step.","tokens_in":44122,"tokens_out":3258,"would_cite":true,"duration_ms":32574,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["81P45","46L10","81P40"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that any synchronous game that robustly self-tests a perfect symmetric projective maximally entangled (PME) strategy also robustly self-tests that strategy for all finite-dimensional POVM strategies, with only polynomial…","keywords":["robust self-testing","synchronous games","maximally entangled strategies","POVM strategies","spectral gap","Quantum Low Degree Test","von Neumann algebras","device-independent certification"],"falsifier":"Construct a finite-dimensional synchronous game $G$ and a family of POVM strategies with winning probability at least $1-1/n$ such that $G$ is $\\kappa$-PME-robust with $\\kappa(\\epsilon)\\to 0$, but the best local dilation distance from these strategies to the ideal PME strategy is bounded below by a positive constant; such an example would contradict Theorem 1.2 and Corollary 5.5.","tokens_in":43023,"feed_emoji":"🔐","tokens_out":11795,"duration_ms":119672,"temperature":0.7,"pith_summary":"Robust self-testing allows a referee to certify that two untrustworthy players are using a specific quantum strategy from their answers alone, but proofs are much easier if one assumes the players use symmetric projective maximally entangled (PME) strategies, an assumption no real device satisfies. The paper answers the question of whether this restriction can be lifted: for every synchronous game, a robust self-test that works for PME strategies automatically works for general finite-dimensional POVM strategies, with a robustness function that degrades only polynomially. The lifting is quantitative and independent of the number of questions and answers; it depends only on the synchronicity parameter $\\beta$ that measures how often the referee checks consistency. As an application, the $1/2$-synchronised Quantum Low Degree Test robustly self-tests the $k$-qubit maximally entangled state together with the Pauli group, with robustness polynomial in $\\log k$ and in the winning-probability gap.","feed_headline":"PME self-tests lift to all quantum strategies","feed_subtitle":"For synchronous games, certification under maximally entangled strategies extends to all POVM strategies with only polynomial loss.","key_machinery":"The load-bearing object is the game polynomial $T_{G,S} = \\mathbb{E}_{(x,y)\\sim\\nu}\\sum_{a,b}D(a,b|x,y)\\,A^x_a\\otimes B^y_b$ and its spectral gap, the difference between its largest and second-largest eigenvalues. Theorem 4.6 converts a spectral gap $\\alpha$ into a local dilation bound $O\\bigl(\\alpha^{-1/2}\\sqrt{(c\\,\\mathrm{id}+\\kappa)(2\\epsilon+\\mathrm{poly}(\\delta))}\\bigr)$ for every near-perfect strategy, and Theorem 5.4 proves that PME-robustness prevents $\\alpha$ from being arbitrarily small. Around this sits the von Neumann-algebraic reformulation: every PME strategy is a triple $(M,\\tau,\\mathcal A)$ of a tracial von Neumann algebra, its trace, and POVMs, and the resulting von Neumann distance between PME strategies is equivalent, up to constants, to the standard local-dilation distance. Vidick's approximate decomposition theorem is used to pass from an arbitrary near-perfect projective strategy to an average of PME strategies, which is where the decomposition of the reduced density matrix into spectral projections enters.","core_discovery":"Theorem 1.2 is the central claim: let $G$ be any synchronous game, and suppose $G$ $\\kappa$-robustly self-tests a perfect PME strategy $\\tilde S$ for PME strategies. Then $G$ $\\kappa'$-robustly self-tests $\\tilde S$ for general POVM strategies, with $\\kappa'$ polynomially related to $\\kappa$. Corollary 5.5 makes this quantitative: $\\kappa'(\\epsilon) \\le C_1 \\sqrt{(\\mathrm{id}+\\kappa)(C_2(\\epsilon/\\beta)^{\\zeta_1})} \\big/ \\bigl[\\beta \\, ((\\mathrm{id}+\\kappa^2)^{-1}(C_3))^{\\zeta_2}\\bigr]$, with universal constants. The proof has three parts: a von Neumann-algebraic distance for PME strategies that is equivalent to the standard Hilbert-space distance; a theorem showing that a PME-robust self-test whose game polynomial has spectral gap is automatically a robust self-test for general strategies; and a theorem showing that PME-robustness itself forces a polynomial lower bound on that spectral gap. Applied to the Quantum Low Degree Test, the paper computes the spectral gap of the game polynomial for a code of relative distance $d$ to be exactly $d/2$, which yields the robust qubit test.","pith_inferences":["The paper restricts to finite-dimensional strategies in Section 2.1, so the abstract's 'all strategies' should be read as all finite-dimensional POVM strategies; extending the lifting to infinite-dimensional strategies would require new arguments and is the first open direction suggested by the proof.","The lifting has the flavour of a meta-theorem: because the constants are universal, future robust self-tests for synchronous games can be proved in the convenient PME/tracial setting and upgraded automatically, which may change how such proofs are written.","The explicit spectral gap $d/2$ for the Quantum Low Degree Test suggests a general recipe: compute the spectral gap of the game polynomial for other code-based synchronous games, and the same machinery will convert a PME-level test into a full test with a quantitative bound.","The generalised notion of a $(\\kappa,\\hat\\nu)$-robust self-test, with different distributions for winning and for distance, may be useful beyond this paper, because it allows certifying only the questions that matter while other questions are present only to enforce the strategy's structure."],"forward_implications":["Any synchronous game that has been proved to be a robust self-test only under PME strategies is now, by Theorem 1.2, a robust self-test for all finite-dimensional POVM strategies, so PME-based self-testing results become physically meaningful.","The quantitative loss in the lifting is polynomial and independent of the size of the game; only the synchronicity parameter $\\beta$ enters, so the upgrade does not get harder as the number of questions and answers grows.","The Quantum Low Degree Test, after applying a $1/2$-synchronicity test, robustly self-tests the $k$-qubit maximally entangled state together with a generating set of Pauli operators, with robustness at most $\\mathrm{poly}(\\log k)\\cdot\\mathrm{poly}(\\epsilon)$.","Because Theorem 5.4 shows PME-robustness automatically gives a spectral gap, the spectral-gap condition in Theorem 4.6 is not an extra assumption; it is available for every PME-robust synchronous self-test.","The exact gap computation $d/2$ for the Quantum Low Degree Test ties the quality of the qubit test directly to the relative distance of the underlying linear code, so better codes would directly improve the test."],"supporting_citations":[{"why":"supplies Theorem 2.12, the approximate convex decomposition of near-perfect projective strategies into PME strategies that the first lifting step relies on.","marker":"[Vid22]"},{"why":"gives the POVM-to-PVM orthogonalization bound used in Lemma 3.4 to move from an almost-synchronous ME strategy to a nearby PME strategy.","marker":"[dlS22a]"},{"why":"provides the Quantum Low Degree Test and the PME-qubit test framework that the paper applies the lifting to.","marker":"[CVY23]"},{"why":"establishes the correspondence between synchronous correlations and tracial C*-algebras, the basis for the PME/von Neumann picture.","marker":"[PSSTW16]"},{"why":"the preceding operator-algebraic self-testing result that showed projectivity can be lifted for synchronous games, which this paper extends by removing maximal entanglement.","marker":"[PSZZ24]"},{"why":"supplies the matrix majorisation theorem used in the spectral-gap lower bound of Theorem 5.4.","marker":"[Bha97]"},{"why":"used in Remark 6.2 to identify the spectral-gap computation for the Quantum Low Degree Test game polynomial.","marker":"[dlS22b]"}],"fun_headline_variants":["Synchronous games self-test without entanglement assumption","Robust self-testing drops maximal entanglement requirement","Quantum self-tests work for all strategies, not just PME","No more PME assumption: robust self-tests for any strategy","Lifting PME: synchronous game self-tests go universal"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument only covers finite-dimensional strategies; the dimension estimate, the decomposition into PME strategies, and the spectral-gap reasoning all use finite-dimensionality in essential ways, so the advertised 'all strategies' is established only for finite-dimensional POVM strategies.","fun_headline_variants_meta":{"raw":{"variants":["Synchronous games self-test without entanglement assumption","Robust self-testing drops maximal entanglement requirement","Quantum self-tests work for all strategies, not just PME","No more PME assumption: robust self-tests for any strategy","Lifting PME: synchronous game self-tests go universal"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000736,"raw_usage":{"total_tokens":3314,"prompt_tokens":992,"completion_tokens":2322,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":608,"completion_tokens_details":{"reasoning_tokens":2241}},"tokens_in":608,"tokens_out":2322,"duration_ms":18021,"temperature":1.0,"reasoning_tokens":2241,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T22:51:39.318582+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct a finite-dimensional synchronous game $G$ and a family of POVM strategies with winning probability at least $1-1/n$ such that $G$ is $\\kappa$-PME-robust with $\\kappa(\\epsilon)\\to 0$, but the best local dilation distance from these strategies to the ideal PME strategy is bounded below by a positive constant; such an example would contradict Theorem 1.2 and Corollary 5.5.","supporting_citations":[],"review_version":1}