Lifting the maximally-entangledness assumption in robust self-testing for synchronous games

Matthijs Vernooij; Yuming Zhao

REVIEW 2 major objections 4 minor 6 references

Lifting the maximally-entangledness assumption in robust self-testing for synchronous games

T0 review · 2 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read 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…

desk verdict Strong quantitative lifting theorem with a real but likely patchable gap in the spectral decomposition step. read the letter →

arxiv 2505.05994 v1 pith:BMJLSMNG submitted 2025-05-09 quant-ph math.OA

Matthijs Vernooij , Yuming Zhao This is my paper · ORCID

classification quant-phmath.OA MSC 81P4546L1081P40

keywords robustself-testingsynchronousgamesmaximallyentangledstrategiesPOVMspectralgapQuantumLowDegreeTestvonNeumannalgebrasdevice-independentcertification

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

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the 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.

What'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.

The 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.

Who 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.

Referee Report

2 major / 4 minor

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.

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 (2)

[Theorem 4.2 and Claim 4.3] 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.
[Lemma 5.2] 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.

minor comments (4)

[Theorem 4.2 setup] 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.
[Abstract and Section 2.1] 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.
[Lemma 2.9] 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.
[Theorem 5.4 proof] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the lifting theorem is a genuine conditional derivation from the PME-robustness assumption, using external theorems and no fitted input renamed as prediction.

full rationale

The derivation chain is not circular. The input is a PME-robust self-test for a synchronous game, and the output is a general POVM-robust self-test; the proofs never assume the target conclusion. Lemma 3.4 applies the PME-robustness assumption to a PME strategy obtained by orthogonalizing an almost synchronous ME strategy, and then transfers the dilation to the original ME strategy. Theorem 4.2 uses Vidick's decomposition (Theorem 2.12) to split a general strategy into near-optimal PME pieces and uses the PME self-test to build isometries; Theorem 4.6 adds the spectral gap of the game polynomial; Theorem 5.4 derives a lower bound on that spectral gap from the PME self-test and the dimension estimate in Lemma 5.1, rather than assuming the bound. No parameter is fitted to a subset of data and then renamed a prediction, and no claimed prediction reduces by construction to the definition of an input. The self-citations to [Zha24], [PSZZ24], and [BMZ24] are contextual or otherwise not load-bearing; in particular, the footnote citing [Zha24] explicitly says those results do not establish the quantitative relationship needed here. The flagged issue that min(Lambda) may fail to exist because spectral projections are only right-continuous is a technical well-definedness gap in the construction, not circularity: it does not make the theorem's conclusion equal to its hypothesis. The paper also states its finite-dimensional restriction explicitly in Section 2.1, which weakens the abstract's 'all strategies' wording but is an assumption, not a circular move. Overall the argument is self-contained conditional on the cited external results from Vidick, de la Salle, Chapman--Vidick--Yuen, and standard matrix analysis.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters are fitted. Constants such as 1700 and 24 are explicit universal constants derived in proofs, not tuned to data. The listed axioms are the external theorems and the finite-dimensional domain assumption that the central proof leans on.

assumptions (5)

domain assumption All strategies are finite-dimensional (Section 2.1: 'we assume all strategies employ finite-dimensional systems').
Central theorem and proofs use finite-dimensional Hilbert spaces, Schmidt decomposition, dimension inequalities in Lemma 5.1, and finite-dimensional von Neumann algebras; infinite-dimensional strategies are excluded.
standard math Vidick's decomposition theorem (Theorem 2.12) for near-perfect projective strategies into approximate PME strategies.
External result from [Vid22] used in Lemmas 2.9, 5.2 and Theorem 4.2; the paper relies on explicit intermediate bounds from its proof.
standard math de la Salle's orthogonalization theorem [dlS22a, Theorem 1.2] used in Lemma 3.4.
Converts POVM elements to nearby PVMs in trace norm; external published theorem.
domain assumption The Quantum Low Degree Test is a (k,kappa)-CPME-qubit test with specified parameters, and the underlying code has distance d.
From [CVY23] and the Reed-Muller code construction, used in Section 2.3 and Theorem 6.1.
standard math Standard functional analysis tools: GNS construction, polar decomposition, Birkhoff-von Neumann theorem, rearrangement inequality, and tracial von Neumann algebra facts.
Background used in the appendix proofs and throughout the main text.

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Pith. "Pith review of Lifting the maximally-entangledness assumption in robust self-testing for synchronous games." pith.science (2026). https://pith.science/paper/BMJLSMNG

@misc{pith2026250505994,
  author       = {Pith},
  title        = {Pith review of: Lifting the maximally-entangledness assumption in robust self-testing for synchronous games},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BMJLSMNG}},
  note         = {Machine review of arXiv:2505.05994}
}

abstract

Robust self-testing in non-local games allows a classical referee to certify that two untrustworthy players are able to perform a specific quantum strategy up to high precision. Proving robust self-testing results becomes significantly easier when one restricts the allowed strategies to symmetric projective maximally entangled (PME) strategies, which allow natural descriptions in terms of tracial von Neumann algebras. This has been exploited in the celebrated MIP*=RE paper and related articles to prove robust self-testing results for synchronous games when restricting to PME strategies. However, the PME assumptions are not physical, so these results need to be upgraded to make them physically relevant. In this work, we do just that: we prove that any perfect synchronous game which is a robust self-test when restricted to PME strategies, is in fact a robust self-test for all strategies. We then apply our result to the Quantum Low Degree Test to find an efficient $n$-qubit test.

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Reference graph

Works this paper leans on

6 extracted references · 3 canonical work pages

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

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Over- lapping Qubits

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The Computational Advantage of MIP ∗ Vanishes in the Presence of Noise

isbn: 9780387972459. REFERENCES 49 [Don+24] Yangjing Dong, Honghao Fu, Anand Natarajan, Minglong Qin, Haochen Xu, and Penghui Yao. “The Computational Advantage of MIP ∗ Vanishes in the Presence of Noise”. In: 39th Computational Complexity Conference (CCC 2024). Ed. by Rahul Santhanam. Vol. 300. Leibniz International Pro- ceedings in Informatics (LIPIcs). ...

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