Skip to main content

Involutive Markov categories and the quantum de Finetti theorem

Antonio Lorenzin ( University of Innsbruck (Austria) )

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.


  1. 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
  2. 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
  3. 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
  4. A. J. Parzygnat. Inverses, disintegrations, and Bayesian inversion in quantum Markov categories. Submitted, 2020. arXiv:2001.08375



Share this: