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

TL;DR

This paper proves that a recent compiler for multipartite non-local games maintains quantum soundness in the asymptotic limit, using operator algebra techniques to relate correlations to quantum commuting strategies.

Contribution

It extends quantum soundness proof to all multipartite games, employing new operator algebra methods and characterizations of no-signalling strategies.

Findings

Quantum correlations correspond to quantum commuting strategies asymptotically.

The proof generalizes previous bipartite results to multipartite scenarios.

Introduces new operator algebra tools, including universal C*-algebras and a chain rule for Radon-Nikodym derivatives.

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.