Quantifying Causal Influences in the Presence of a Quantum Common Cause

Quantum mechanics challenges our intuition on the cause-effect relations in nature. Some fundamental concepts, including Reichenbach’s common cause principle or the notion of local realism, have to be reconsidered. Traditionally, this is witnessed by the violation of a Bell inequality. But are Bell inequalities the only signature of the incompatibility between quantum correlations and causality theory? Motivated by this question, we introduce a general framework able to estimate causal influences between two variables, without the need of interventions and irrespectively of the classical, quantum, or even postquantum nature of a common cause. In particular, by considering the simplest instrumental scenario—for which violation of Bell inequalities is not possible—we show that every pure bipartite entangled state violates the classical bounds on causal influence, thus, answering in negative to the posed question and opening a new venue to explore the role of causality within quantum theory.

Figure 1

Directed acyclic graphs depicting causal structures: (left) Instrumental scenario and (right) Bell scenario. In the quantum model, we consider, here, the classical common source, described by a random variable $\mathrm{\Lambda }$, is replaced by a quantum (potentially entangled) state ${\rho }_{AB}$.

Figure 2

(left) Violation $v_{\alpha }$ of the classical bound by an entangled two-qubit pure state with parameter $\alpha$; and violation $v_{\varphi }$ as a function of the angle $\varphi$ between projective measurements of Bob. The dashed lines show that optimal states and measurements are different from the maximal entangled state ($\alpha =(\pi /4)$) and measurements in mutually unbiased bases ($\varphi =(\pi /4)$). (right) Regions with nonzero lower bounds on CACE (Ref. [35]), QACE [Eq. (11)], and NACE [Eq. (12)].