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

TL;DR

This paper extends causal inference rules to quantum systems with entanglement, revealing limitations in classical causal assumptions when applied to indefinite causal order quantum processes, exemplified by a quantum switch circuit.

Contribution

It introduces a CP-do(C)-calculus that unifies causal modeling across classical, definite, and indefinite quantum causal regimes, and shows Rule 2's failure in indefinite causal order.

Findings

Rule 2 of do-calculus fails with indefinite causal order

Demonstration using a three-qubit quantum switch circuit

Provides a unified causal modeling framework for classical and quantum regimes

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.