Discrimination between quantum common causes and quantum causality

Abstract

In classic cases, Reichenbach's principle implies that discriminating between common causes and causality is unprincipled since the discriminative results essentially depend on the selection of possible conditional variables. For some typical quantum cases, K.Reid $et$ $al$. \href{https://www.nature.com/articles/nphys3266}{[Nat. Phys. 11, 414 (2015)]} presented the statistic $C$ which can effectively discriminate quantum common causes and quantum causality over two quantum random variables (i.e., qubits) and which only uses measurement information about these two variables. In this paper, we formalize general quantum common causes and general quantum causality. Based on the formal representation, we further investigate their decidability via the statistic $C$ in general quantum cases. We demonstrate that (i) $C \in \left[ { - 1,\frac{1}{{27}}} \right]$ if two qubits are influenced by quantum common causes; (ii) $C \in \left[ { - \frac{1}{{27}},1} \right]$ if the relation between two qubits is quantum causality; (iii) a geometric picture can illuminate the geometric interpretation of the probabilistic mixture of quantum common causes and quantum causality. This geometric picture also provides a basic heuristic to develop more complete methods for discriminating the cases corresponding to $C \in \left[ { - \frac{1}{{27}},\frac{1}{{27}}} \right]$. Our results demonstrate that quantum common causes and quantum causality can be discriminated in a considerable scope.