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Description
We generalize Pearl´s definition of causal influence from classical random variables and probability distributions to the quantum world. For two systems $A$, $B$, embedded in a common environment $C$ that might represent quantum fields and mediate an interaction between $A$ and $B$, we find the necessary and sufficient condition on a unitary U that jointly propagates $A,B,C$ for a causal influence of $A$ on $B$ and vice versa. We then introduce an easily computable measure of the causal influence and use it to study the causal influence of different quantum gates, the mutuality of the causal influence, and quantum superpositions of different causal orders. We study the distribution of causal influence over an ensemble of Haar-distributed unitaries $U$ and calculate analytically the expected value of the causal influence. Applied to two two-level atoms that dipole-interact with a thermal bath of electromagnetic waves, we find that the space-time dependence of causal influence almost perfectly reproduces the one of reservoir-induced entanglement. [1] Llorenç Escolà-Farràs and Daniel Braun, Phys. Rev. A, 106, 062415 (2022)
