Quantifying causal influence in quantum mechanics

We extend Pearl's definition of causal influence to the quantum domain, where two quantum systems $A, B$ with finite-dimensional Hilbert space are embedded in a common environment $C$ and propagated with a joint unitary $U$. For a finite-dimensional Hilbert space $C$, we find the necessary and sufficient condition on $U$ for a causal influence of $A$ on $B$ and vice versa. We introduce an easily computable measure of the causal influence and use it to study the causal influence of different quantum gates, its mutuality, and quantum superpositions of different causal orders. For two two-level atoms dipole-interacting with a thermal bath of electromagnetic waves, the space-time dependence of causal influence almost perfectly reproduces the one of reservoir-induced entanglement.