Uniqueness of the Petz retrodiction functor

Prove that every retrodiction functor on the category of von Neumann algebras equipped with normal faithful states and state-preserving normal completely positive unital maps coincides with the Petz retrodiction functor, thereby uniquely characterizing Bayesian inference and Petz recovery by the axioms in Definition 4.2.

Background

The paper establishes that the Petz recovery map defines a retrodiction functor on the category of von Neumann algebras with normal faithful states and state-preserving normal completely positive unital maps. This functor satisfies recovery, identity preservation, compositionality, tensoriality, extension of inversion, and involutivity.

The authors explicitly leave unresolved whether these categorical axioms uniquely determine the Petz retrodiction functor. Establishing uniqueness would show that classical Bayesian inference and its quantum generalization are structural necessities rather than merely selected inference algorithms.

References

Motivated by the results of Ref., which showed that a variety of other endofunctors (such as the rotated Petz recovery map and its averaged variants~\cites{JRSWW16,JRSWW18}) are not retrodiction functors, we state the following conjecture about the uniqueness of retrodiction. If $\mathscr{R}$ is a retrodiction functor on $\mathbf{Q}$, then $\mathscr{R}$ coincides with the Petz retrodiction functor. In other words, Petz retrodiction is uniquely characterized by the axioms in Definition~\ref{defn:retrodiction}. In particular, Bayesian inference is characterized by these same axioms.

— Bayesian inference and retrodiction for faithful states on von Neumann algebras  (2608.20001 - Karmakar et al., 20 Aug 2026) in Conjecture in Section 9, immediately following Definition 9.1 (page number unavailable)