Observational-Interventional Bell Inequalities

Generalizations of Bell's theorem, particularly within quantum networks, are now being analyzed through the causal inference lens. However, the exploration of interventions, a central concept in causality theory, remains significantly unexplored. In this work we give an initial step in this direction, by analyzing the instrumental scenario and proposing novel hybrid Bell inequalities integrating observational and interventional data. Focusing on binary outcomes with any number of inputs, we obtain the complete characterization of the observational-interventional polytope, equivalent to a Hardy-like Bell inequality albeit describing a distinct quantum experiment. To illustrate its applications, we show a significant enhancement regarding threshold detection efficiencies for quantum violations also showing the use of these hybrid approach in quantum steering scenarios.