Involutive Markov categories and the quantum de Finetti theorem

Tobias Fritz; Antonio Lorenzin

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arXiv · 2312.09666

Involutive Markov categories and the quantum de Finetti theorem

Tobias Fritz·Antonio Lorenzin

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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.

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Tobias Fritz, Antonio Lorenzin. 2026-08-22. Involutive Markov categories and the quantum de Finetti theorem. https://arxiv.org/abs/2312.09666

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