Recoverability and Irreversible Information Loss in Physical Inference: Lessons From Quantum Measurement and Climate Reanalysis

ABSTRACT Quantum Mechanics (QM) and Climate Science (CS) confront the epistemic problem of inferring an unobservable state from incomplete, indirect, and context‐dependent measurements. Although their physics differ profoundly (non‐commutative algebra vs. multi‐scale fluid dynamics), their knowledge updates can be compared through a variational, information‐projection view of Bayesian updating under observational constraints. In QM, this principle is encoded by the Lüders rule, which admits an entropic‐projection interpretation (minimizing the change in under the measurement constraint ). In CS, it is realized by data assimilation (DA) under multi‐scale limitations and strong information loss due to sparse, noisy proxies and temporal smoothing in paleoclimate records. In both domains, updates can be cast as minimizing an information‐theoretic distance subject to the likelihood imposed by data . We propose correspondences between the quantum instrument and climate analysis (assimilation) update operators, between POVM elements and proxy‐system likelihoods , and between quantum recoverability (Petz‐type recovery ) and backward information propagation in smoothing and reanalysis. This framework targets the diagnosis of irreversible information loss in measurement and inference.