Indistinguishability of causal relations from limited marginals

We investigate the possibility of distinguishing among different causal relations starting from a limited set of marginals. Our main tool is the notion of adhesivity, that is, the extension of probability or entropies defined only on subsets of variables, which provides additional independence constraints among them. Our results provide a criterion for recognizing which causal structures are indistinguishable when only limited marginal information is accessible. Furthermore, the existence of such extensions greatly simplifies the characterization of a marginal scenario, a result that facilitates the derivation of Bell inequalities both in the probabilistic and entropic frameworks, and the identification of marginal scenarios where classical, quantum, and postquantum probabilities coincide.

Figure 1

Examples of graphs and hypergraphs. (a) A tree graph $\mathcal{T}$ where $\{B,A,D\}$ is a path. (b) A graph $\mathcal{G}$ where $\{A,C,B,A\}$ is a loop and $\{A,B,C\}$ and $\{A,C,D\}$ are cliques. $B$ and $D$ are separated by $\{A,C\}$. (c) A directed graph where $\{A,C,B,A\}$ is not a directed path, because direction of $(A,B)$ is not respected. This graph does not contain loops; i.e., it is a directed acyclic graph (DAG). $B$ and $D$ are as well separated by $\{A,C\}$, but $\left\{C\right\}$ is a minimal separator. (d) A hypergraph $\mathcal{H}$ where nodes $B$ and $D$ are separated by $\{A,C\}$.

Figure 2

Example of graphs and hypergraphs. (a) A cyclic hypergraph $\mathcal{H}$. (b) A 2-section graph $\mathcal{G}={\left[\mathcal{H}\right]}_2$ of the hypergraph. (c) A clique hypergraph ${\mathcal{H}}^{\prime }$ of $\mathcal{G}$. Notice that ${\mathcal{H}}^{\prime }\ne \mathcal{H}$. In fact, ${\mathcal{H}}^{\prime }$ is an extension of $\mathcal{H}$.

Figure 3

Examples of different Bayesian networks. (a) A DAG representing a Markov chain $X\to Y\to Z$ implying the CI $(X\perp Z|Y)$. (b) A DAG where the variable $Y$ is a common parent of $X$ and $Z$, once more implying the CI $(X\perp Z|Y)$. (c) A DAG where the variables $X$ and $Z$ have a common child $Y$. In this case $(X\perp Z)$, but $(X\neg \perp Z|Y)$.

Figure 4

Hypergraph of the marginal scenario associated with a Bell-CHSH experiment. The observed probabilities correspond to the marginal for $\{A_x,B_y\}$, for $x,y=1,2$.

Figure 5

Procedure to compute the minimal hypergraph $\mathcal{T}$ extending the marginal scenario $\mathcal{M}$ for the CHSH scenario. (a) Initial marginal scenario hypergraph corresponding to the CHSH case. (b) 2-section ${\left[\mathcal{M}\right]}_2$ of the original hypergraph. (c) Triangulation of the 2-section graph. (d) Clique hypergraph $\mathcal{T}$ of the triangulation. $\mathcal{T}$ extends $\mathcal{M}$.

Figure 6

Marginal scenario hypergraph (a) and (b) its 2-section.

Figure 7

Marginal scenario (a) and a clique hypergraph (b).

Figure 8

Example of causal structures [(a), (c)] and marginal scenarios [(b), (d)].

Figure 9

Causal structure (a) and marginal scenario (b) of classical case of information causality. $X_0,X_1$ are random inputs for Alice, $Y_0,Y_1$ are guesses for Bob, and $M$ is a message which Alice sends to Bob.