Causal Interventions Beyond Time: A CP-do(C)-Calculus for Indefinite Quantum Order

Abstract: We reformulate Pearl's three rules of do-calculus in the language of completely positive (CP) trace-preserving maps, thereby extending them to quantum systems with entanglement. We prove that Rule~2 fails whenever the underlying process admits indefinite causal order, and we demonstrate this failure in a three-qubit ``quantum switch'' circuit. Our analysis clarifies why the classical notions of surgical intervention, faithfulness, and counterfactual dependence must be revised in quantum information science. The CP-do($C$)-calculus introduced here provides a common syntax for causal modelling across classical, definite-order quantum, and indefinite-order quantum regimes.