Deterministic Underlying States are incompatible with a Counterfactual account of Lüders' rule

In this work, we show that a counterfactual account of L"uders’ rule – which we argue is naturally implied by the mathematical structure of the rule itself – rules out underlying-state models of quantum mechanics (a type of hidden-variable model, typically used in the contextuality and nonlocality literature, where quantum states are treated as probability measures over ``better-defined states’'). This incompatibility arises because the counterfactual update requires ontological models to update their states according to conditional probability, which in turn establishes an equivalence between compatibility and the existence of such models.