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.

翻译：非局域游戏是区分经典世界与量子世界中可能关联的有力工具。Kalai等人（STOC'23）提出了一种编译器，可将多方非局域游戏转化为与单一证明者的交互式协议，其通过密码学工具消除了参与者物理分离的假设。虽然该构造对所有多方游戏已证明具有量子完备性与经典可靠性，但量子可靠性仅在双方游戏的特例中已知。本文通过证明渐近情形下可生成的任何关联均对应于量子交换策略，证实了Kalai等人的编译器确实对所有多方编译非局域游戏实现了量子可靠性。我们的证明运用了算子代数理论的技术，并依赖于将多方情形下的序列操作无信号策略刻画为量子交换算子策略，从而推广了若干先前结果。在此过程中，我们构建了序列投影值测度的通用C*-代数，并证明了C*-代数上完全正映射的Radon-Nikodym导数的新链式法则，该结果可能具有独立的理论价值。