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arXiv:2505.05994 · Quantum

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

Matthijs Vernooij, Yuming Zhao

Published Oct 1, 2026• 0 citations
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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 M I P ∗ = R E 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.

DOI
10.22331/q-2026-10-01-2222

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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 M I P ∗ = R E 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.
We develop an abstract operator-algebraic characterization of robust self-testing for synchronous correlations and games. Specifically, we show that a synchronous correlation is a robust self-test if and only if there is a unique state on an appropriate C∗C^*C∗-algebra that "implements" the correlation. Extending this result, we prove that a synchronous game is a robust self-test if and only if its associated C∗C^*C∗-algebra admits a unique amenable tracial state. This framework allows us to establish that all synchronous correlations and games that serve as commuting operator self-tests for finite-dimensional strategies are also robust self-tests. As an application, we recover sufficient conditions for linear constraint system games to exhibit robust self-testing. We also demonstrate the existence of a synchronous nonlocal game that is a robust self-test but not a commuting operator self-test, showing that these notions are not equivalent.
We show that exact self-testing of a quantum correlation does not imply robust self-testing. Mančinska and Schmidt previously established such a separation for non-local games, but their obstruction to robustness arises from two distinct optimal correlations, with the self-tested one remaining robust. We establish the separation for a single correlation. Our proof is operator-algebraic. The main technical contribution is a general construction that associates a synchronous binary correlation to any finitely presented C*-algebra with unitary generators and real-coefficient relations. The implementing states of the correlation correspond exactly to the tracial states on the algebra. We apply this construction to an algebra with a unique finite-dimensional tracial state and a distinct amenable tracial state. By a characterization established by Zhao and independently by Kar, the resulting correlation is a self-test but not a robust self-test.
We give an operator-algebraic formulation of robust self-testing in terms of states on C*-algebras. We show that a quantum correlation p is a robust self-test only if among all (abstract) states, there is a unique one achieving p. We show that the "if" direction of this statement also holds, provided that p is optimal/perfect for a nonlocal game that has a robust game algebra. This last condition applies to many nonlocal games of interest, including all XOR games, synchronous games, and boolean constrained system (BCS) games. For those nonlocal games with robust game algebras, we prove that self-testing is equivalent to the uniqueness of finite-dimensional tracial states on the associated game algebra, and robust self-testing is equivalent to the uniqueness of amenable tracial states. Applying this tracial-state characterization of self-testing to parallel repetition, we show that a synchronous game is a self-test for perfect quantum strategies if and only if its parallel repeated version is a self-test for perfect quantum strategies. As a proof approach, we give the first quantitative Gower-Hatami theorem that is applicable to C*-algebras. Here "quantitative" means there is a constructive bound on the distance between the approximate representations and exact representations. We also demonstrate how this quantitative Gowers-Hatami theorem can be used to calculate the explicit robustness function of a self-test.
We construct a nonlocal game that self-tests a maximally entangled qubit strategy among pure full-Schmidt-rank projective strategies, but admits an inequivalent optimum using one nonprojective measurement. This resolves a conjecture of Baptista et al. on imposing full rank and projectivity simultaneously. The construction combines CHSH with an auxiliary game GGG that forces deterministic answers under these assumptions and admits a trine POVM optimum without them. We classify the optimal correlations of GGG as a line segment parametrized by the tracial states of C⊕M2(C)\mathbb{C}\oplus M_2(\mathbb{C})C⊕M2​(C). Projectivity on the state support removes the matrix summand, whereas projective dilations with nonzero nonabort probability have a nontracial local state whose zero left ideal is not two-sided. Finally, a family in local dimension six shows that the full-rank PVM self-test is not robust.
We reduce Lalonde's pseudo-telepathy inner product game from six to five dimensions, keeping its four Alice questions and three Bob questions and reducing each answer alphabet to five. We prove that the reduced game self-tests its nonmaximally entangled state and all 35 measurement effects. The self-test is also robust: for a strategy winning with probability at least 1−ε1-\varepsilon1−ε we obtain dext≤min⁡{1,221ε}d_{\mathrm{ext}}\le\min\{1,2^{21}\sqrt{\varepsilon}\}dext​≤min{1,221ε​}, where dext=1−Ξd_{\mathrm{ext}}=\sqrt{1-Ξ}dext​=1−Ξ​ and ΞΞΞ is the optimal squared fidelity with the reference state under local extraction channels. A compactness argument then gives simultaneous qualitative robustness for all measurement actions on the state under common local isometries. As a consequence of the self-test, no maximally entangled state of any finite dimension can win the game perfectly.
Self-testing results allow us to infer the underlying quantum mechanical description of states and measurements from classical outputs produced by non-communicating parties. The standard definition of self-testing does not apply in situations when there are two or more inequivalent optimal strategies. To address this, we introduce the notion of self-testing convex combinations of reference strategies, which is a generalisation of self-testing to multiple strategies. We show that the Glued Magic Square game [Quantum 4 (2020), p. 346] self-tests a convex combination of two inequivalent strategies. As a corollary, we obtain that the Glued Magic square game self-tests two EPR pairs thus answering an open question from [Quantum 4 (2020), p. 346]. Our self-test is robust and extends to natural generalisations of the Glued Magic Square game.