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arXiv:2302.11783 · New Journal of Physics

A semantics for counterfactuals in quantum causal models

Ardra Kooderi Suresh, Markus Frembs, Eric G. Cavalcanti

Published Jun 1, 2026• 1 citation
arXivHTMLPDF

Abstract

Abstract We introduce a formalism for the evaluation of counterfactual queries in the framework of quantum causal models, generalising Pearl’s semantics for counterfactuals in classical causal models, thus completing the last rung in the quantum analogue of Pearl’s ‘ladder of causation’. To this end, we define a suitable extension of Pearl’s notion of a ‘structural causal model’, which we denote analogously by ‘quantum structural causal model (QSM)’ , and a corresponding extension of Pearl’s three-step procedure of abduction, action, and prediction. We show that every classical (probabilistic) structural causal model can be extended to a QSM, and prove that counterfactual queries that can be formulated within a classical structural causal model agree with their corresponding queries in the quantum extension—but the latter is more expressive. Counterfactuals in quantum causal models come in different forms: we distinguish between active and passive counterfactual queries, depending on whether or not an intervention is to be performed in the action step. This is in contrast to the classical case, where counterfactuals are always interpreted in the active sense. Another distinctive feature of our formalism is that it breaks the connection between causal and counterfactual dependence that exists in the classical case: quantum counterfactuals allow for counterfactual dependence without causal dependence . This distinction between classical and quantum causal models may shed light on how the latter can reproduce quantum correlations that violate Bell inequalities while being faithful to the relativistic causal structure.

DOI
10.1088/1367-2630/ae78f6

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Abstract We introduce a formalism for the evaluation of counterfactual queries in the framework of quantum causal models, generalising Pearl’s semantics for counterfactuals in classical causal models, thus completing the last rung in the quantum analogue of Pearl’s ‘ladder of causation’. To this end, we define a suitable extension of Pearl’s notion of a ‘structural causal model’, which we denote analogously by ‘quantum structural causal model (QSM)’ , and a corresponding extension of Pearl’s three-step procedure of abduction, action, and prediction. We show that every classical (probabilistic) structural causal model can be extended to a QSM, and prove that counterfactual queries that can be formulated within a classical structural causal model agree with their corresponding queries in the quantum extension—but the latter is more expressive. Counterfactuals in quantum causal models come in different forms: we distinguish between active and passive counterfactual queries, depending on whether or not an intervention is to be performed in the action step. This is in contrast to the classical case, where counterfactuals are always interpreted in the active sense. Another distinctive feature of our formalism is that it breaks the connection between causal and counterfactual dependence that exists in the classical case: quantum counterfactuals allow for counterfactual dependence without causal dependence . This distinction between classical and quantum causal models may shed light on how the latter can reproduce quantum correlations that violate Bell inequalities while being faithful to the relativistic causal structure.
Costa and Shrapnel ([2016]) have recently proposed an interventionist theory of quantum causation. The formalism generalizes the classical methods of Pearl ([2000]) and allows for the discovery of quantum causal structure via localized interventions. Classical causal structure is presented as a special case of this more general framework. I introduce the account and consider whether this formalism provides a causal explanation for the Bell correlations. 1 Introduction2 Classical Causal Models3 What’s the (Quantum) Problem?4 Quantum Causal Models: Why Bother?5 Markov Quantum Causal Models6 The Bell Experiment7 Bell’s Objections
We show that quantum oracles provide an advantage over classical oracles for answering classical counterfactual questions in causal models, or equivalently, for identifying unknown causal parameters such as distributions over functional dependences. In structural causal models with discrete classical variables, observational data and even ideal interventions generally fail to answer all counterfactual questions, since different causal parameters can reproduce the same observational and interventional data while disagreeing on counterfactuals. Using a simple binary example, we demonstrate that if the classical variables of interest are encoded in quantum systems and the causal dependence among them is encoded in a quantum oracle, coherently querying the oracle enables the identification of all causal parameters -- hence all classical counterfactuals. We generalize this to arbitrary finite cardinalities and prove that coherent probing 1) allows the identification of all two-way joint counterfactuals p(Y_x=y, Y_{x'}=y'), which is not possible with any number of queries to a classical oracle, and 2) provides tighter bounds on higher-order multi-way counterfactuals than with a classical oracle. This work can also be viewed as an extension to traditional quantum oracle problems such as Deutsch--Jozsa to identifying more causal parameters beyond just, e.g., whether a function is constant or balanced. Finally, we raise the question of whether this quantum advantage relies on uniquely non-classical features like contextuality. We provide some evidence against this by showing that in the binary case, oracles in some classically-explainable theories like Spekkens' toy theory also give rise to a counterfactual identifiability advantage over strictly classical oracles.
Identifying causal relations plays an indispensable role in science, economics, medicine and everyday life. Interventionist causation associates the discovery of such relations with the possibility of manipulation or intervention. A causes B, if manipulating A in the right way can bring about changes in B. The main technical tool, the causal model, can be seen as a device for telling us how and why certain actions are more effective than others. Such models have found application in almost every scientific discipline. However, despite several decades of sophisticated analysis and a wealth of technical results, interventionist causal methods are known to fail in the quantum case. The quantum domain, with its peculiar, counterintuitive features, is widely considered an inhospitable environment in which to apply our usual causal discovery tools. This causal skepticism presents us with a puzzle. Physicists are now able to manipulate and control quantum systems, and the design and construction of quantum technology relies on an ability to identify when one strategy will work and another will fail. From the perspective of interventionist causation, it seems that a formal account of quantum causal modelling ought to be possible. The main contribution of this thesis is to provide an extension of interventionist methodology to the quantum case. In Chapter 1, I introduce interventionist causation from the perspective of two giants of the field: Judea Pearl and James Woodward. The Causal Markov Condition (CMC) and Faithfulness assumptions are presented, and I draw particular attention to the discovery of casual structure. I touch on some of the well-known problems with interventionist causation per se. In Chapter 2, I put this methodology to work in a quantum mechanical context. Following tradition, we examine one of the Bell experiments in order to determine how the classical interventionist framework fails. Noting that the usual response is to demand that one drops either the CMC or Faithfulness, I suggest that neither approach is particularly satisfactory and argue for an alternative. I discuss some reasons why the interventionist is forced to pursue this different path. In Chapter 3, I motivate the possibility of an interventionist account of quantum causation. I first look at how physicists represent quantum technology in order to make interventionist and counterfactual inferences: via the use of quantum circuit diagrams. Using three examples I demonstrate the manner in which these diagrams can be used to facilitate causal inference. In Chapter 4, I examine some recent attempts at characterising quantum causal models due to work from the quantum foundations community. I present three such accounts and explain why I endorse a fourth, the focus of the next chapter. In Chapter 5, I present a new framework that allows for the discovery of interventionist quantum causal structure. I finish the thesis with a summary and some suggestions for further work.
Abstract The existence of non-local correlations between outcomes of measurements in quantum entangled systems strongly suggests that we are dealing with some form of causation here. An assessment of this conjecture in the context of the collapse interpretation of quantum mechanics is the primary goal of this paper. Following the counterfactual approach to causation, I argue that the details of the underlying causal mechanism which could explain the non-local correlations in entangled states strongly depend on the adopted semantics for counterfactuals. Several relativistically-invariant interpretations of spatiotemporal counterfactual conditionals are discussed, and the corresponding causal stories describing interactions between parts of an entangled system are evaluated. It is observed that the most controversial feature of the postulated causal connections is not so much their non-local character as a peculiar type of circularity that affects them.
This paper presents a framework for quantum causal modeling based on the interpretation of causality as a relation between an observer's probability assignments to hypothetical or counterfactual experiments. The framework is based on the principle of ``causal sufficiency'': that it should be possible to make inferences about interventions using only the probabilities from a single ``reference experiment'' plus causal structure in the form of a directed acyclic graph. This leads to several interesting results: we find that quantum measurements deserve a special status distinct from interventions, and that a special rule is needed for making inferences about what would happen if they are not performed (``unmeasurements''). One natural candidate for this rule is found to be an equation of importance to the quantum Bayesianism interpretation of quantum mechanics. We find that the causal structure of quantum systems must have a ``layered'' structure, and that the model can naturally be made symmetric under reversal of the causal arrows.