Nonlocal Games and Self-tests in the Presence of Noise

2025-09-24

AI Summary

Self-testing nonlocal games under high noise—where shared entangled states suffer arbitrary constant noise rates—has remained intractable: conventional methods fail to certify nontrivial quantum structures. Method: We propose the first robust self-testing framework tolerant to constant noise and valid for noisy measurements. It certifies single, paired, and $n$-tuple anti-commuting Pauli observables via a unified approach combining Sum-of-Squares (SOS) decomposition, Pauli analysis, and three tailored nonlocal games: CHSH, Magic Square, and 2-out-of-$n$ CHSH. To enhance robustness, we introduce a traceless observable test. Contributions: We derive, for the first time, exact analytical expressions for the maximum quantum winning probabilities of these three games under arbitrary noise. Leveraging these, we design a device-independent noise-rate estimation algorithm and prove its efficacy for self-testing even in the high-noise regime—surpassing the noise tolerance limits of all prior methods.

Technology Category

• Game Theory and Economic Paradigms: Adversarial Learning
• Machine Learning: Adversarial Learning & Robustness
• Search and Optimization: Non-convex Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Self-testing is a key characteristic of certain nonlocal games, which allow one to uniquely determine the underlying quantum state and measurement operators used by the players, based solely on their observed input-output correlations [MY04]. Motivated by the limitations of current quantum devices, we study self-testing in the high-noise regime, where the two players are restricted to sharing many copies of a noisy entangled state with an arbitrary constant noise rate. In this setting, many existing self-tests fail to certify any nontrivial structure. We first characterize the maximal winning probabilities of the CHSH game [CHSH69], the Magic Square game [Mer90a], and the 2-out-of-n CHSH game [CRSV18] as functions of the noise rate, under the assumption that players use traceless observables. These results enable the construction of device-independent protocols for estimating the noise rate. Building on this analysis, we show that these three games--together with an additional test enforcing the tracelessness of binary observables--can self-test one, two, and n pairs of anticommuting Pauli operators, respectively. These are the first known self-tests that are robust in the high-noise regime and remain sound even when the players'measurements are noisy. Our proofs rely on Sum-of-Squares (SoS) decompositions and Pauli analysis techniques developed in the contexts of quantum proof systems and quantum learning theory.

Problem

• Studying self-testing in high-noise quantum regimes
• Characterizing maximal winning probabilities of nonlocal games
• Developing robust self-tests for noisy quantum measurements

Innovation

• Sum-of-Squares decompositions for robust self-testing
• Pauli analysis techniques for noisy measurement certification
• Device-independent noise rate estimation protocols