Synchronous games with *-isomorphic game algebras [article]

Samuel J. Harris
<span title="2021-09-10">2021</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
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
more &raquo; ... 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.
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2109.04859v1">arXiv:2109.04859v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/hp4444m5wbejjarbrizki3oh34">fatcat:hp4444m5wbejjarbrizki3oh34</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20210930220830/https://arxiv.org/pdf/2109.04859v1.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/5b/6a/5b6ac2ed02f389c698b94fc3e6fb0380cfb61a1c.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2109.04859v1" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>