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

Matthijs Vernooij, Yuming Zhao

TL;DR

This work addresses the challenge of removing the PME restriction in robust self-testing for synchronous non-local games. By formulating PME strategies in a tracial von Neumann algebra framework and proving a distance equivalence to the Hilbert-space setting, the authors show that PME-robust self-testing implies robust self-testing for all POVMs with a polynomially related robustness κ', depending only on the game’s synchronicity. The analysis ties robustness to a spectral gap of the ideal strategy, establishing a polynomial lower bound Δ ≥ C1 β ((id+κ^2)^{-1}(C2))^ζ, and applies these ideas to the Quantum Low Degree Test, where Δ = d/2, yielding an explicit, efficient n-qubit test. The results significantly advance device-independent certification by removing unphysical PME assumptions and providing concrete, scalable tests with provable robustness. Collectively, the paper strengthens connections between operator-algebra techniques, spectral-gap methods, and quantum self-testing in the MIP*=RE landscape.

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.

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

TL;DR

This work addresses the challenge of removing the PME restriction in robust self-testing for synchronous non-local games. By formulating PME strategies in a tracial von Neumann algebra framework and proving a distance equivalence to the Hilbert-space setting, the authors show that PME-robust self-testing implies robust self-testing for all POVMs with a polynomially related robustness κ', depending only on the game’s synchronicity. The analysis ties robustness to a spectral gap of the ideal strategy, establishing a polynomial lower bound Δ ≥ C1 β ((id+κ^2)^{-1}(C2))^ζ, and applies these ideas to the Quantum Low Degree Test, where Δ = d/2, yielding an explicit, efficient n-qubit test. The results significantly advance device-independent certification by removing unphysical PME assumptions and providing concrete, scalable tests with provable robustness. Collectively, the paper strengthens connections between operator-algebra techniques, spectral-gap methods, and quantum self-testing in the MIP*=RE landscape.

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 -qubit test.
Paper Structure (13 sections, 29 theorems, 259 equations)

This paper contains 13 sections, 29 theorems, 259 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Proof approach and technical contributions
  4. Acknowledgement
  5. Preliminaries
  6. Non-local games and strategies
  7. Almost synchronous correlations
  8. Quantum low degree test
  9. Self-testing in the von Neumann algebra picture
  10. Robust self-testing based on the spectral gap of the game polynomial
  11. Spectral gap for robust synchronous self-tests
  12. Spectral gap of the Quantum Low Degree Test
  13. Proof of Lemma \ref{['lem:PME-robust-self-testing-vNA']}

Key Result

Theorem 1.2

Let $G$ be a synchronous game. If $G$$\kappa$-robustly self-tests a perfect quantum strategy $\tilde{\mathcal{S}}$ for PME strategies, then $G$$\kappa'$-robustly self-tests $\tilde{\mathcal{S}}$ for general POVM strategies, where $\kappa'$ is polynomially related to $\kappa$.

Theorems & Definitions (70)

  • Theorem 1.2
  • Theorem 1.3: Precise statement in \ref{['cor:QLDT']}
  • Definition 2.1: Local dilation
  • Remark 2.2
  • Definition 2.3: Self-testing
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • ...and 60 more