title: Counterfactuals in Classical and Quantum Causal Models
Creators: name: Kooderi Suresh, Ardra
Publisher: name: Griffith University
Year: 2024
Subjects: subject: quantum causal models
subject: counterfactuals
subject: classical split-node causal models

description: Causal models find applications in medical reasoning and research, decision-making, weather prediction, policy-making etc. However, the classical causal models formalism fails to explain correlations observed in quantum experiments while satisfying desirable principles, such as maintaining a causal structure compatible with the theory of relativity and the possibility of discovering the underlying causal structure via observations and interventions. This motivated the introduction of quantum causal models. Ever since, the field of quantum causal models has been an expanding one, with different frameworks being proposed and novel results. However, a framework to explain and evaluate counterfactual probabilities in quantum causal models has been missing so far. The major aim of this thesis is to fill that gap. This gap becomes important once we note that counterfactuals form the third and highest level in the causal hierarchy introduced by Judea Pearl for classical causal models, which he calls the ladder of causation. The first two levels, of association and intervention, have already been given analogous treatments in quantum causal models. In this thesis, we complete the picture by providing a detailed analysis of counterfactual reasoning in quantum causal models and proposing a framework for formulating and evaluating well-defined quantum counterfactual queries. Building upon existing frameworks of quantum causal models, we define a quantum structural causal model, analogous to a (classical) structural causal model, which is the class of models encoding the required structure for questions at level 3 of Pearl's hierarchy. In our framework, a "standard quantum counterfactual query" can be evaluated using a three-step procedure analogous to that introduced by Pearl to evaluate counterfactuals in classical causal models. We show that the classical case is returned as a special case of the quantum framework, but the quantum semantics is richer. For example, unlike in the classical case, where all counterfactual statements are either true or false when given the complete description of a structural causal model, the quantum case allows in general only for counterfactual probabilities. More remarkably, we find that in the quantum case, it is possible to have counterfactual dependence without causal dependence, a feature that cannot occur in Pearl's classical formalism. In order to better delineate the uniquely quantum features from what could be artefacts of the different structure represented by the causal graphs of the quantum formalism, we also develop a semantics for counterfactuals in classical split-node causal models. These are classical causal models that resemble a quantum causal model in that a node is, instead of a single random variable, associated with input and output systems, where an intervention can be performed via an instrument. This analysis shows that some "quantum-like" features such as nontrivial counterfactual probabilities can arise when non-ideal classical instruments are used (but not when ideal instruments are used), whereas some features like counterfactual dependence without causal dependence do not arise simply as artefacts of the split-node structure. The semantics presented in the thesis contributes to the literature on both quantum and classical causal models. It adds to the hierarchical description of causal questions as illustrated in Pearl's ladder of causation and fills the aforementioned gap in explaining counterfactuals in quantum causal models. The framework for counterfactuals in classical split-node causal models also provides not only important insights into the differences between quantum and classical frameworks but could also be potentially useful in classical applications where the split-node structure, with instruments being left outside of the model, may be a natural description.