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Sufficiency of the counterfactual account of Lüders’ rule to rule out ontological models of quantum mechanics

Tezzin A, Amaral B, Hance JR

Published Nov 11, 2025• 0 citations
Publisher

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 L"uders’ 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.

DOI
10.57711/9jfw-y885

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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 L"uders’ 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.
Ontological models, as used in the generalized contextuality literature, play a central role in current research on quantum foundations, providing a framework for defining classicality, constructing classical analogs 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 renders predictions of sequential measurements order-independent. This implies that ontological models, even contextual ones, must either act differently than what we would expect given (this, typically implicitly assumed account of) quantum state update rule, or cannot model quantum behavior.
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—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.
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—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.
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—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.
Since the famous debates of Einstein & Bohr, physicists have argued about the nature of the quantum state. Is it best thought of as describing the-way-things-are out there in the world, or merely as a description of our knowledge of such things? In more recent years, this has been termed the ψ-epistemic/ψ-ontic debate, and that distinction has been given a mathematical definition within the ontological models formalism. This formalism is a framework for describing a large class of interpretations of quantum mechanics. Here we show that consideration of sequential measurements, and the fact that the quantum state changes during measurements, has been a neglected topic in this area as it places nontrivial restrictions on the structure of ψ-epistemic ontological models. \n \nWe do this by finding a general restriction on the structure of ψ-epistemic models, although not one that is strong enough to rule them out categorically. We then apply this restriction to all of the known examples of ψ-epistemic ontological models and show that they can’t represent sequential measurements. We also present a new version of the ontological models formalism, based on hidden Markov models, which we develop briefly and describe how it may be useful for either proving a no-go theorem (i.e. ‘ψ-epistemic models can’t exist’) or for learning more about the structure of ψ-epistemic models and how to construct them.
Realist interpretations of quantum mechanics presuppose the existence of elements of reality that are independent of the actions used to reveal them. Such a view is challenged by several no-go theorems that show quantum correlations cannot be explained by non-contextual ontological models, where physical properties are assumed to exist prior to and independently of the act of measurement. However, all such contextuality proofs assume a traditional notion of causal structure, where causal influence flows from past to future according to ordinary dynamical laws. This leaves open the question of whether the apparent contextuality of quantum mechanics is simply the signature of some exotic causal structure, where the future might affect the past or distant systems might get correlated due to non-local constraints. Here we show that quantum predictions require a deeper form of contextuality: even allowing for arbitrary causal structure, no model can explain quantum correlations from non-contextual ontological properties of the world, be they initial states, dynamical laws, or global constraints.