Conditional Distributions for Quantum Systems

Conditional distributions, as defined by the Markov category framework, are studied in setting of matrix algebras (quantum systems). Their construction linear unital maps obtained via a categorical Bayesian inversion procedure. Simple criteria establishing when such positive obtained. Several examples provided, including standard EPR scenario, where correlations reproduced purely compositional (categorical) manner. A comparison between Bayes map, Petz recovery and Leifer-Spekkens acausal belief propagation is illustrating some similarities key differences.