Inverses, disintegrations, and Bayesian inversion in quantum Markov categories
arXiv:2001.08375
Abstract
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 -algebraic notions of almost everywhere (a.e.) equivalence. As a consequence, we prove many results including a universal no-broadcasting theorem for S-positive categories, a generalized Fisher--Neyman factorization theorem for a.e. modular categories, a relationship between error correcting codes and disintegrations, and the relationship between Bayesian inversion and Umegaki's non-commutative sufficiency.
arXiv commentsubstantial changes, 91 pages, comments welcome!
References in corpus (16)
- Noncommuting mixed states cannot be broadcast
- A generalized no-broadcasting theorem
- A Unified and Generalized Approach to Quantum Error Correction
- Towards a Formulation of Quantum Theory as a Causally Neutral Theory of Bayesian Inference
- Reversing quantum dynamics with near-optimal quantum and classical fidelity
- A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics
- Recoverability in quantum information theory
- Disintegration and Bayesian Inversion via String Diagrams
- Quantum Dynamics as an analog of Conditional Probability
- Picturing classical and quantum Bayesian inference
- Why initial system-environment correlations do not imply the failure of complete positivity: a causal perspective
- Anti- (Conjugate) Linearity
- From Kleisli Categories to Commutative C*-algebras: Probabilistic Gelfand Duality
- From observables and states to Hilbert space and back: a 2-categorical adjunction
- A non-commutative Bayes' theorem
- Quantum Programs as Kleisli Maps