arXiv 2109.048592021-09-10quant-phmath.OA

具有$*$ -同构对策代数的同步对策

Synchronous games with $*$-isomorphic game algebras

Samuel J. Harris

备注：: 20 pages

摘要：

我们建立了几个强等价的同步非局部游戏，在这个意义上，相应的游戏代数是$*$ -同构。我们首先证明了$n$输入和$k$输出上的任何同步博弈的博弈代数是$*$ -同构于$nk$输入和$nk$输出上的相关双同步博弈的博弈代数。结果表明，存在问题集和答案集相等的双同步博弈，其最优策略只存在于量子交换模型中，而不存在于量子近似模型中。此外，我们展示了一个双同步游戏的$20$的问题和$20$的答案，有一个非零的游戏代数，但没有获胜的交换策略，解决问题的V.I. Paulsen和M.拉哈曼我们还展示了一个$*$ -同构之间的任何同步游戏代数与$n$问题和$k>3$答案和一个同步游戏代数与$n(k-2)$问题和$3$答案。

英文摘要：

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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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