Tobias Fritz, Tomáš Gonda, Paolo Perrone, Eigil Fjeldgren Rischel, Representable Markov categories and comparison of statistical experiments in categorical probability, Theoretical Computer Science 961, 2023. (arXiv:2010.07416)

Noé Ensarguet, Paolo Perrone, Categorical probability spaces, ergodic decompositions, and transitions to equilibrium, arXiv:2310.04267

For the quantum case:

Sam Staton and Ned Summers, Quantum de Finetti Theorems as Categorical Limits, and Limits of State Spaces of $C^\ast$-algebras, Proceedings of Quantum Physics and Logic, 2022. (arXiv)

Tobias Fritz and Antonio Lorenzin, Involutive Markov categories and the quantum de Finetti theorem, 2023. (arXiv)