De Finetti theorems tell us that if we expect the likelihood of outcomes to be independent of their order, then these sequences of outcomes could be equivalently generated by drawing an experiment at random from a distribution, and repeating it over and over. In particular, the quantum de Finetti theorem says that exchangeable sequences of quantum states are always represented by distributions over a single state produced over and over. The main result of this paper is that this quantum de Finetti construction has a universal property as a categorical limit. This allows us to pass canonically between categorical treatments of finite dimensional quantum theory and the infinite dimensional. The treatment here is through understanding properties of (co)limits with respect to the contravariant functor which takes a C*-algebra describing a physical system to its convex, compact space of states, and through discussion of the Radon probability monad. We also show that the same categorical analysis also justifies a continuous de Finetti theorem for classical probability.

翻译：德菲内蒂定理告诉我们，如果我们预期结果的似然性与顺序无关，那么这些结果序列等价于从一个分布中随机抽取实验并反复重复而生成。特别地，量子德菲内蒂定理指出，量子态的可交换序列总能用重复生成的单个态上的分布来表示。本文的主要结果是，该量子德菲内蒂构造具有作为范畴极限的泛性质。这使得我们能在有限维量子理论与无限维量子理论的范畴刻画之间建立正则转换。本文通过理解（余）极限相对于逆变函子的性质（该函子将描述物理系统的C*-代数映射到其紧凸态空间），以及通过讨论拉东概率单子来实现这一目标。我们还证明，相同的范畴分析也证明了经典概率的连续德菲内蒂定理。