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.

Disclosure

“and, we have S12 = J12 ∆12 , and on the other hand, we have 1 1 1/2 1/2 S12 = J1 ∆12 ⊗ J2 ∆22 = (J1 ⊗ J2 )(∆1 ⊗ ∆2 ). AI usage declaration. Microsoft 365 Copilot and Google Gemini were used to search for references, to generate BibTex data, to learn background information, for exploratory mathemat- ical assistance, and for proofreading. All results generated by this method were checked. All of the writing was done s”

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Classification

Brainstorming or outlining

Multiplier

2

Verified

2026-08-22T18:11:19.025821Z

Structural counts