Causal influence versus signaling for interacting quantum channels

A causal relation between quantum agents, say Alice and Bob, is necessarily mediated by an interaction. Modeling the latter as a reversible quantum channel, an intervention of Alice can have causal influence on Bob's system, modifying correlations between Alice and Bob's systems. Causal influence between quantum systems necessarily allows for signaling. Here we prove a mismatch between causal influence and signaling via direct computation of the two quantities for the Cnot gate. Finally we show a continuity theorem for causal effects of unitary channels: a channel has small causal influence if it allows for small signaling.