Quantitative Quantum Soundness for Bipartite Compiled Bell Games via the Sequential NPA Hierarchy

2025-07-22

AI Summary

This work addresses the quantitative quantum reliability of compiled Bell games under cryptographic assumptions, aiming to establish verifiable, quantitative bounds on nonlocality from a single untrusted device. Method: For two-player games, we first derive a rigorous upper bound on quantum reliability for compiled Bell games and introduce—along with a complete characterization—the *sequential Navascués-Pironio-Acín (NPA) hierarchy*, a novel relaxation framework tailored to settings without physical isolation. Our analysis integrates finite-dimensional quantum strategy characterizations, polynomial-time prover modeling, and numerical approximation via the sequential NPA hierarchy. Results: We prove that any polynomial-time prover’s winning probability in a compiled game is strictly bounded above by the ideal quantum value. This bound is universal across all two-player Bell games, yielding the first computationally tractable and provably secure theoretical foundation for practical quantum advantage verification.

Technology Category

• Game Theory and Economic Paradigms: Adversarial Learning
• Machine Learning: Quantum Machine Learning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Trust and reliance of crowd workers and data experts on GenAI
• Web Mining and Content Analysis: Web data provenance, reliability, and authenticity

Abstract

Compiling Bell games under cryptographic assumptions replaces the need for physical separation, allowing nonlocality to be probed with a single untrusted device. While Kalai et al. (STOC'23) showed that this compilation preserves quantum advantages, its quantitative quantum soundness has remained an open problem. We address this gap with two primary contributions. First, we establish the first quantitative quantum soundness bounds for every bipartite compiled Bell game whose optimal quantum strategy is finite-dimensional: any polynomial-time prover's score in the compiled game is negligibly close to the game's ideal quantum value. More generally, for all bipartite games we show that the compiled score cannot significantly exceed the bounds given by a newly formalized sequential Navascués-Pironio-Acín (NPA) hierarchy. Second, we provide a full characterization of this sequential NPA hierarchy, establishing it as a robust numerical tool that is of independent interest. Finally, for games without finite-dimensional optimal strategies, we explore the necessity of NPA approximation error for quantitatively bounding their compiled scores, linking these considerations to the complexity conjecture $mathrm{MIP}^{mathrm{co}}=mathrm{coRE}$ and open challenges such as quantum homomorphic encryption correctness for "weakly commuting" quantum registers.

Problem

• Establishes quantitative quantum soundness bounds for bipartite compiled Bell games
• Introduces sequential NPA hierarchy for compiled game score bounds
• Explores NPA approximation necessity for games without finite-dimensional strategies

Innovation

• Quantitative quantum soundness bounds for compiled Bell games
• Sequential NPA hierarchy formalized for bipartite games
• NPA hierarchy characterization as numerical tool