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.
Keywords
Cite
@article{arxiv.2312.09666,
title = {Involutive Markov categories and the quantum de Finetti theorem},
author = {Tobias Fritz and Antonio Lorenzin},
journal= {arXiv preprint arXiv:2312.09666},
year = {2026}
}
Comments
52 pages. This version improves the clarity of some analytical arguments (in particular, the proofs of Proposition 3.40 and Theorem 4.40)