Quantum violation of average causal effects in multiple measurement settings

Estimating causal relations from observed correlations is a central content of science. Although a comprehensive mathematical framework has been developed to identify cause and effect, it is well known that such methods and techniques are not applicable to quantum systems due to Bell's theorem. Generally, the incompatibility between quantum correlation and classical causal theory is illustrated by the violation of Bell inequality. Gachechiladze provides a new method called the violation of lower bound of average causal effect (ACE) to witness the incompatibility. We consider a new lower bound of ACE derived by Cao in multiple measurement settings. We show that there are always pure entangled states and incompatible projective measurements that can generate correlations that violate this new classical lower bound. In Gachechiladze's work, the measurement settings are binary, while the measurement settings in our paper are K-ary (K >= 2) and we give more general conclusions. When K = 2, the same result as in Gachechiladze's work can be obtained.