Sufficiency of the counterfactual account of Lüders' rule to rule out ontological models of quantum mechanics
Alisson Tezzin, Bárbara Amaral, Jonte R. Hance
TL;DR
This work probes whether ontological models that underpin generalized contextuality can faithfully reproduce Lüders' state-update rule when interpreted counterfactually. By formalizing state update in ontological terms as conditional probability, the authors prove that an ontological model can reproduce Lüders' rule only if all measured observables are pairwise compatible; hence, quantum incompatibility yields a fundamental obstruction to such hidden-variable representations. The main results extend prior deterministic findings to stochastic models and rely on Bayes’ rule and the Kolmogorov extension theorem to show that sequential measurements of incompatible observables necessarily conflict with conditional-probability updates. The work thereby challenges the sufficiency of generalized contextuality as a universal notion of classicality and clarifies the precise way in which quantum state updates resist classical ontological encodings, with implications for finite vs infinite-dimensional systems and potential links to nonlocality and epistemic interpretations. Overall, the paper provides a rigorous foundation tying quantum state update, measurement incompatibility, and ontological models together, highlighting a key no-go for hidden-variable accounts within the standard quantum framework.
Abstract
Ontological models, as used in the generalised contextuality literature, play a central role in current research on quantum foundations, providing a framework for defining classicality, constructing classical analogues of key quantum phenomena, and examining the ontology of quantum states. In this work, we show that a counterfactual account of Lüders' rule -- which we argue is naturally implied by the mathematical structure of the rule itself -- renders such models inherently incompatible with the quantum formalism. This incompatibility arises because the counterfactual update requires ontological models to update their states according to conditional probability, which in turn which in turn renders predictions of sequential measurements order-independent. This implies that ontological models, even contextual ones, must either act differently to what we would expect given (this, typically implicitly-assumed account of) quantum state update rule, or cannot model quantum behaviour.