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Markov categories have recently emerged as a powerful high-level frameworkfor probability theory and theoretical statistics. Here we study a quantumversion of this concept, called involutive Markov categories. First, we showthat these are equivalent to Parzygnat's quantum Markov categories but arguethat they are simpler to work with. Our main examples of involutive Markovcategories involve C*-algebras (of any dimension) as objects and completelypositive unital maps as morphisms in the picture of interest. Second, we provea quantum de Finetti theorem for both the minimal and the maximal C*-tensornorms, and we develop a categorical description of such quantum de Finettitheorems which amounts to a universal property of state spaces.

Involutive Markov categories and the quantum de Finetti theorem