Involutive Markov categories and the quantum de Finetti theorem
Tobias Fritz, Antonio Lorenzin
TL;DR
The paper develops a quantum generalization of Markov categories, introducing involutive Markov categories (ICD-cats) to accommodate infinite-dimensional pre-C*-algebras and completely positive unital maps. It builds a robust toolkit—deterministic ICD-functors, cofree ICD-categories, strictification, and the notions of classicality, compatibility, and pictures—to isolate physically meaningful probabilistic morphisms and to formalize state spaces as universal objects. A central achievement is a quantum de Finetti theorem valid for both the minimal and maximal tensor norms, expressed via state-space representability and a universal property that connects exchangeable morphisms to state spaces. The framework provides a unified, diagrammatic approach to quantum probability, enabling a categorical account of de Finetti representations and a rigorous treatment of infinite-dimensional quantum structures, while clarifying how classical and quantum components interact through classical representability and Kolmogorov-type products.
Abstract
Markov categories have recently emerged as a powerful high-level framework for probability theory and theoretical statistics. Here we study a quantum version of this concept, called involutive Markov categories. These are equivalent to Parzygnat's quantum Markov categories, but we argue that they offer a simpler and more practical approach. Our main examples of involutive Markov categories have pre-C*-algebras, including infinite-dimensional ones, as objects, together with completely positive unital maps as morphisms in the picture of interest. In this context, we prove a quantum de Finetti theorem for both the minimal and the maximal C*-tensor norms, and we develop a categorical description of such quantum de Finetti theorems which amounts to a universal property of state spaces.