Inverses, disintegrations, and Bayesian inversion in quantum Markov categories

Understand

We introduce quantum Markov categories as a structure that refines and extends a synthetic approach to probability theory and information theory so that it includes quantum probability and quantum information theory.

• In this broader context, we analyze three successively more general notions of reversibility and statistical inference: ordinary inverses, disintegrations, and Bayesian inverses.
• We prove that each one is a strictly special instance of the latter for certain subcategories, providing a categorical foundation for Bayesian inversion as a generalization of reversing a process.
• We unify the categorical and $C^*$-algebraic notions of almost everywhere (a.e.) equivalence.