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Lüders conditionalization: Conditional probability, transition probability, and updating in quantum probability theory

John Earman

Published Apr 15, 2019• 0 citations
OpenAlex

Abstract

Guerra Bobo (2013) has questioned whether Lüders conditionalization for quantum probabilities supplies a notion of conditional probability worthy of the name. I agree in large part with Guerra Bobo's critique; indeed, I show how her critique can be sharpened. But while valuable in itself, the main virtue of the critique derives from the fact that it prompts questions about the nature of quantum probabilities that engage some of the more important and contentious issues in the foundations of QM. I show how understanding the role of Lüders conditionalization in quantum probability theory helps to illuminate these issues.

Related papers

On Quantum Conditional Probability

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We argue that quantum theory does not allow for a generalization of the notion of classical conditional probability by showing that the probability defined by the Lüders rule, standardly interpreted in the literature as the quantum-mechanical conditionalization rule, cannot be interpreted as such.

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Discussion: Quantum Logic and The Lüders Rule

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In a recent paper, Michael Friedman and Hilary Putnam argued that the Lüders rule is ad hoc from the point of view of the Copenhagen interpretation but that it receives a natural explanation within realist quantum logic as a probability conditionalization rule. Geoffrey Hellman maintains that quantum logic cannot give a non-circular explanation of the rule, while Jeffrey Bub argues that the rule is not ad hoc within the Copenhagen interpretation. As I see it, all four are wrong. Given that there is to be a projection postulate, there are at least two natural arguments which the Copenhagen advocate can offer on behalf of the Lüders rule, contrary to Friedman and Putnam. However, the argument which Bub offers is not a good one. At the same time, contrary to Hellman, quantum logic really does provide an explanation of the Lüders rule, and one which is superior to that of the Copenhagen account, since it provides an understanding of why there should be a projection postulate at all.

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Incompatibility of deterministic underlying states with a counterfactual account of Lüders' rule

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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.

•Nov 1, 2025•0 citations
Guerra Bobo (2013) has questioned whether Lüders conditionalization for quantum probabilities supplies a notion of conditional probability worthy of the name. I agree in large part with Guerra Bobo's critique; indeed, I show how her critique can be sharpened. But while valuable in itself, the main virtue of the critique derives from the fact that it prompts questions about the nature of quantum probabilities that engage some of the more important and contentious issues in the foundations of QM. I show how understanding the role of Lüders conditionalization in quantum probability theory helps to illuminate these issues.
In a recent paper, Michael Friedman and Hilary Putnam argued that the Lüders rule is ad hoc from the point of view of the Copenhagen interpretation but that it receives a natural explanation within realist quantum logic as a probability conditionalization rule. Geoffrey Hellman maintains that quantum logic cannot give a non-circular explanation of the rule, while Jeffrey Bub argues that the rule is not ad hoc within the Copenhagen interpretation. As I see it, all four are wrong. Given that there is to be a projection postulate, there are at least two natural arguments which the Copenhagen advocate can offer on behalf of the Lüders rule, contrary to Friedman and Putnam. However, the argument which Bub offers is not a good one. At the same time, contrary to Hellman, quantum logic really does provide an explanation of the Lüders rule, and one which is superior to that of the Copenhagen account, since it provides an understanding of why there should be a projection postulate at all.
A core tenet of Bayesian epistemology is that rational agents update by conditionalization.Accuracy arguments in favour of this norm are well-known.Meanwhile, scholars working in quantum probability and quantum state estimation have proposed multiple updating rules, all of which look prima facie like analogues of Bayesian conditionalization.The most common are Lüders conditionalization and Bayesian mean estimation (BME).Some authors also endorse a lesser-known alternative that we call retrodiction.We show how one can view Lüders and BME as complementary rules, and we give expected accuracy and accuracy dominance arguments for both.By contrast, we find that retrodiction is accuracy-dominated, at least on many measures of accuracy.Bayesian norms do not immediately carry over to this setting.(Here and in what follows, we assume that there are objective quantum probabilities, and that quantum systems have true, objective quantum states.The idea that there are both subjective and objective probabilities, and we can discuss their relationship, is sometimes known as Lewisian dualism (Wallace [2012]).)While it is unclear how to translate standard Bayesian norms into this quantum operatortheoretic setting, we can draw a natural analogy.Usually, the Bayesian represents both credences and states of the world as real-valued functions over an algebra of propositions; we can view credences as 'estimates' of the truth values assigned by a given state. 1 In quantum theory, one represents states with bounded, linear operators on a Hilbert space.It turns out that one can also use such operators to encode estimates of quantum states.We adopt this paradigm to construct natural analogues of the accuracy arguments, affording novel Bayesian justifications for several quantum updating rules.It is controversial whether such a non-classical approach to probability is necessary once we fix an appropriate solution to the measurement problem (Loewer, Healey [1994, 2020]).However, even if it is not necessary, this alternate setting is immensely popular.Experimenters typically work within the framework of Hilbert-space operators, as this framework houses most approaches to quantum state estimation (that is, the reconstruction of quantum states based on measurement outcomes; see Jones, Blume-Kohout, Granade et al. [1991Granade et al. [ , 2010Granade et al. [ , 2016]]).So, the status of Bayesian norms in this setting seems well worth exploring.Interestingly, authors have proposed multiple updating norms in this framework, all of which look prima facie like analogues of Bayesian conditionalization:
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.
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.