Bayesian retrodiction of quantum supermaps

The Petz map has been established as a quantum version of the Bayes' rule. It unifies the conceptual belief update rule of a quantum state observed after a forward quantum process, and the operational reverse process that brings the final state to a recovered state equal to the updated belief, counteracting the forward process. Here, we study a higher-order generalization of the quantum Bayes' rule by considering a quantum process undergoing a quantum supermap. For a few families of initial beliefs, we show that a similar unification is possible -- the rules to update the belief of quantum channels can be implemented via a "reverse" quantum supermap, which we call the retrodiction supermap, allowing for applications such as error correction in quantum cloud computing. Analytical solutions are provided for those families, while a recipe for arbitrary initial beliefs is yet to be found.