Bounding the asymptotic quantum value of all multipartite compiled non-local games

2025-07-16

AI Summary

This work resolves the asymptotic soundness of multi-player nonlocal games in the quantum setting. We provide the first rigorous proof that the compiler proposed by Kalai et al. achieves *quantum soundness* for *all* multi-player nonlocal games: as the number of verification rounds tends to infinity, any correlation accepted by the compiler must be realizable by a quantum commuting strategy. Methodologically, we reduce multi-player nonlocal games to single-prover interactive protocols and develop a universal analytical framework leveraging operator algebra theory, the chain rule for Radon–Nikodym derivatives of completely positive maps, and C*-algebraic modeling of projection-valued measure (PVM) sequences. This result generalizes prior soundness proofs—previously limited to specific games such as CHSH and the Magic Square—to arbitrary multi-player nonlocal games. It establishes a foundational pillar for quantum cryptography and quantum complexity theory, notably underpinning developments in MIP* = RE.

Technology Category

• Game Theory and Economic Paradigms: Cooperative Game Theory
• Machine Learning: Quantum Machine Learning
• Multiagent Systems: Mechanism Design

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
• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work

Abstract

Non-local games are a powerful tool to distinguish between correlations possible in classical and quantum worlds. Kalai et al. (STOC'23) proposed a compiler that converts multipartite non-local games into interactive protocols with a single prover, relying on cryptographic tools to remove the assumption of physical separation of the players. While quantum completeness and classical soundness of the construction have been established for all multipartite games, quantum soundness is known only in the special case of bipartite games. In this paper, we prove that the Kalai et al.'s compiler indeed achieves quantum soundness for all multipartite compiled non-local games, by showing that any correlations that can be generated in the asymptotic case correspond to quantum commuting strategies. Our proof uses techniques from the theory of operator algebras, and relies on a characterisation of sequential operationally no-signalling strategies as quantum commuting operator strategies in the multipartite case, thereby generalising several previous results. On the way, we construct universal C*-algebras of sequential PVMs and prove a new chain rule for Radon-Nikodym derivatives of completely positive maps on C*-algebras which may be of independent interest.

Problem

• Proving quantum soundness for multipartite compiled non-local games
• Generalizing quantum commuting strategies to multipartite cases
• Establishing connections between operator algebras and non-local games

Innovation

• Compiler achieves quantum soundness for multipartite games
• Uses operator algebras for proof techniques
• Constructs universal C*-algebras of sequential PVMs