Involutive Markov categories and the quantum de Finetti theorem

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.