Quantum de Finetti Theorems as Categorical Limits, and Limits of State Spaces of C*-algebras

TL;DR

This paper reveals that quantum de Finetti theorems can be understood as categorical limits, providing a universal framework that bridges finite and infinite-dimensional quantum theories and extends to classical probability.

Contribution

It introduces a categorical limit perspective on quantum de Finetti theorems, connecting finite and infinite-dimensional quantum state spaces through (co)limits and functor analysis.

Findings

Quantum de Finetti theorems are characterized as categorical limits.

The framework bridges finite and infinite-dimensional quantum theories.

A continuous de Finetti theorem for classical probability is justified.

Abstract

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.