# Involutive Markov categories and the quantum de Finetti theorem

- 14:00 1st December 2023 ( week 8, Michaelmas Term 2023 )Lecture Theatre B

Categorical probability, and in particular Markov categories, has attracted much interest in recent years. This synthetic approach has proven fruitful in several areas, such as statistics [1], graphical models [2], ergodic theory [3], and more.

Regarding the quantum perspective, Parzygnat introduced the concept of quantum Markov Categories in [4], where he focused on finite-dimensional C*-algebras.

His work explains that, in the quantum perspective, probability is encoded by well-behaved subcategories rather than by the whole quantum Markov Categories.

This talk offers an alternative description of quantum Markov Categories, which we call *involutive Markov categories*, and their subcategories of interest, which we call *pictures*. This view has the advantage of avoiding any distinction between odd and even morphisms.

We then discuss a particular example which comprises infinite-dimensional (pre)-C*-algebras, and argue that the theory of such algebras can benefit from this new synthetic perspective.

Indeed, we are able to provide a meaningful framework for the quantum de Finetti theorem via *representability*.

This concept is also important for classical Markov categories [1].

In particular, representability implies that the category under consideration is a Kleisli category [1,Theorem 3.19].

It turns out that a quantum version of representability can also arise from a well-behaved limit of exchangeable morphisms, which we call the *de Finetti limit*.

We will prove that such limits exist (even for positive unital maps, and for both the maximal and the minimal norm), and thus a quantum de Finetti theorem holds in this setting.

Moreover, this result allows us to describe the deterministic behaviour of the C*-algebra of complex-valued continuous maps from the state space of a C*-algebra.

References:

- T. Fritz, T. Gonda, P. Perrone, and E. F. Rischel.
*Representable Markov categories and comparison of statistical experiments in categorical probability*. Theoretical Computer Science 961(113986), 2023. arXiv:2010.07416 - T. Fritz and A. Klingler.
*The d-separation criterion in categorical probability*. Journal of Machine Learning Research 24(46):1-49, 2023. arXiv:2207.05740 - S. Moss and P. Perrone.
*A category-theoretic proof of the ergodic decomposition theorem*. Ergodic Theory and Dynamical Systems, pp. 1-27, 2023. arXiv:2207.07353 - A. J. Parzygnat.
*Inverses, disintegrations, and Bayesian inversion in quantum Markov categories*. Submitted, 2020. arXiv:2001.08375