Inability of the entropy vector method to certify nonclassicality in linelike causal structures

Bell's theorem shows that our intuitive understanding of causation must be overturned in light of quantum correlations. Nevertheless, quantum mechanics does not permit signaling and hence a notion of cause remains. Understanding this notion is not only important at a fundamental level, but also for technological applications such as key distribution and randomness expansion. It has recently been shown that a useful way to decide which classical causal structures could give rise to a given set of correlations is to use entropy vectors. These are vectors whose components are the entropies of all subsets of the observed variables in the causal structure. The entropy vector method employs causal relationships among the variables to restrict the set of possible entropy vectors. Here, we consider whether the same approach can lead to useful certificates of nonclassicality within a given causal structure. Surprisingly, we find that for a family of causal structures that includes the usual bipartite Bell structure they do not. For all members of this family, no function of the entropies of the observed variables gives such a certificate, in spite of the existence of nonclassical correlations. It is therefore necessary to look beyond entropy vectors to understand cause from a quantum perspective.

Figure 1

The standard causal structure of a bipartite Bell experiment. Here $A,B,X,$ and $Y$ are observed; $A$ and $B$ correspond to input settings and $X$ and $Y$ to outcomes. If the common cause is classical, then the observed correlations satisfy $P_{ABXY}={\sum }_C{P}_A{P}_B{P}_C{P}_{X|AC}{P}_{Y|BC}$. In the case that $A,B,X,$ and $Y$ are binary, the Clauser-Horne-Shimony-Holt inequality [8] can be derived. However, if the common cause is quantum, this inequality can be violated, but Tsirelson's bound must hold instead [21].

Figure 2

Example causal structure. If this is quantum, then the realizable correlations are those formed by measuring states of the form ${\rho }_{C_X{C}_Y{C}_Z}\otimes {\sigma }_{D_Z{D}_W}$ with separate measurements on ${\mathcal{H}}_{C_X},{\mathcal{H}}_{C_Y},{\mathcal{H}}_{C_Z{D}_Z},$ and ${\mathcal{H}}_{D_W}$.

Figure 3

The causal structure $P_n$. The nodes $X_i$ represent observed variables, whereas $C_i$ denote the unobserved classical or quantum systems.

Figure 4

Causal structure ${\tilde{P}}_3$ representing the independencies of the variables in $P_{X_0{X}_{1}Y{Z}_0{Z}_1}^{\prime }$.