The most standard description of symmetries of a mathematical structure produces a group. However, when the definition of this structure is motivated by physics, or information theory, etc., the respective symmetry objects might become more sophisticated: quasigroups, loops, quantum groups, ... In this paper, we introduce and study quantum symmetries of very general categorical structures: operads. Its initial motivation were spaces of probability distributions on finite sets. We also investigate here how structures of quantum information, such as quantum states and some constructions of quantum codes are algebras over operads.
翻译:数学结构对称的最标准描述产生一个组。 但是,当该结构的定义受物理或信息理论等驱动时, 相应的对称对象可能会变得更加复杂: 准组、循环、量子组. 在本文件中,我们介绍并研究非常普通的绝对结构的量的对称性: 剧本。 它最初的动机是有限机组的概率分布空间。 我们还在这里调查量子信息的结构,例如量子状态和量子代码的某些构造,是如何在数子组之上的代数。