Bayesian inference and retrodiction for faithful states on von Neumann algebras

Abstract:Retrodiction is the act of inferring a cause from its effects, the most common example of which is Bayesian inference. Retrodiction can be defined by its structural process-theoretic properties, which are mathematically captured by category theory. This categorical definition of retrodiction has recently been shown to potentially isolate the Petz recovery map as a unique universal candidate for quantum Bayesian inference. This paper extends these results to the infinite-dimensional setting on von Neumann algebras. In the process, we provide a pedagogical review of the Petz recovery map in infinite dimensions and its relation to the more commonly used expression in the finite-dimensional setting. We formalize the open question as to whether these categorical axioms for retrodiction do in fact uniquely characterize the Petz recovery map. If such a characterization holds, this would show that Bayesian inversion and the Petz recovery map are structural necessities and not simply useful algorithms for classical and quantum inference.

Comments are welcome! 30 pages + appendix + bibliography

Operator Algebras (math.OA); Category Theory (math.CT); Quantum Physics (quant-ph)

arXiv:2608.20001 [math.OA]

(or arXiv:2608.20001v1 [math.OA] for this version)

https://doi.org/10.48550/arXiv.2608.20001 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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