Synchronous games with $*$-isomorphic game algebras

TL;DR

This paper explores the algebraic structures of synchronous non-local games, establishing strong equivalences and demonstrating the existence of games with strategies only in the quantum commuting model, highlighting differences between quantum models.

Contribution

It proves $*$-isomorphisms between game algebras of different synchronous and bisynchronous games, and constructs examples distinguishing quantum models.

Findings

Existence of bisynchronous games with strategies only in the quantum commuting model.

Construction of a bisynchronous game with 20 questions and answers with no winning commuting strategy.

$*$-isomorphism between synchronous game algebras with different question and answer counts.

Abstract

We establish several strong equivalences of synchronous non-local games, in the sense that the corresponding game algebras are $*$-isomorphic. We first show that the game algebra of any synchronous game on $n$ inputs and $k$ outputs is $*$-isomorphic to the game algebra of an associated bisynchronous game on $nk$ inputs and $nk$ outputs. As a result, we show that there are bisynchronous games with equal question and answer sets, whose optimal strategies only exist in the quantum commuting model, and not in the quantum approximate model. Moreover, we exhibit a bisynchronous game with $20$ questions and $20$ answers that has a non-zero game algebra, but no winning commuting strategy, resolving a problem of V.I. Paulsen and M. Rahaman. We also exhibit a $*$-isomorphism between any synchronous game algebra with $n$ questions and $k>3$ answers and a synchronous game algebra with $n(k-2)$ questions and $3$ answers.