Bayesian retrodiction of quantum supermaps

TL;DR

This paper extends the quantum Bayes' rule to quantum supermaps, introducing retrodiction supermaps that update beliefs about quantum channels and have applications in quantum error correction.

Contribution

It introduces the concept of retrodiction supermaps as a higher-order generalization of quantum Bayes' rule for quantum processes.

Findings

Retrodiction supermaps can unify belief updates and reverse quantum processes.

Analytical solutions are provided for specific initial beliefs.

Potential applications include improved quantum error correction.

Abstract

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 recovers the final state to match the updated belief, effectively 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 updating the beliefs about quantum channels can be implemented via a "reverse" quantum supermap, termed the retrodiction supermap. The potential applications of retrodiction supermap are demonstrated with examples of improved error correction in quantum cloud computing. Analytical solutions are provided for these families, while a recipe for arbitrary initial beliefs remains an open question.