Synchronous games with $*$-isomorphic game algebras

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.