Graphical Tests of Causality

TL;DR

The paper develops a unified graphical framework to test various causal orders (static, definite, bi-causal) through simple, graph-based inequalities that bound winning probabilities in graphical games. It builds a geometric bridge between single-output correlations and directed graphs, enabling facet-defining tests via polytopes and their lifting to correlation space. A central contribution is the identification of kefalopoda, cycle/fence/Möbius, not-strong, and minimally strong digraph families that yield tractable, polygonal constraints, plus a polynomial-time algorithm for recognizing weakly causal correlations. The work also introduces a projection-based methodology that preserves tractability while capturing key causal features, and it outlines several open questions about the completeness of these tests and extensions beyond the single-output setting. Overall, the approach provides device-independent, graph-theoretic tests of causality with practical implications for analyzing complex multi-party causal structures.

Abstract

Bell inequalities limit the possible observations of non-communicating parties. Here, we present analogous inequalities for any number of communicating parties under the causal constraints of static causal order, definite causal order, and bi-causal order. All derived inequalities are remarkably simple. They correspond to upper bounds on the winning chance in graphical games: Given a specific directed graph over the parties, the parties are challenged to communicate along a randomly chosen arc. In the case of definite causal order, every game that we find is specified by a kefalopoda digraph. Based on this we define weakly causal correlations as those that satisfy all kefalopoda inequalities. We show that the problem of deciding whether some correlations are weakly causal is solvable in polynomial time in the number of parties.

Figures (9)

• Figure 1: (a) Alice and Bob are situated at distant locations without the possibility to communicate and share some resource. Alice computes the outcome $a\in\{0,1\}$ from her measurement setting $x\in\{0,1\}$. Similarly, Bob computes $b\in\{0,1\}$ from $y\in\{0,1\}$. If the resource is a "lookup table," i.e., a classical object such as a book or a shared random variable, then their correlations cannot violate the Bell inequality \ref{['eq:bell']}. In contrast, if the resource is some quantum state, then they may violate that inequality. (b) The predicate of the Bell inequality \ref{['eq:bell']} requires both outcomes to be different in the case $x=y=1$, and equal otherwise. An immediate logical consequence of this is that no "lookup table" exist that satisfies the predicate, i.e., the constraint $a(0)=b(0)=a(1)\neq b(1)=a(0)$ is unsatisfiable abramsky2012, and at most three-out-of-four (in)equalities may be upheld simultaneously.
• Figure 2: Examples of three graphical games of causality: (a) a Möbius digraph to test static causal order, (b) a kefalopoda digraph to test definite causal order, and (c) a minimally strong digraph to test bi-causal order.
• Figure 3: Illustration of the single-output scenario with $n=10$. Each box represents a party. The particular arrangement of the parties in this figure is arbitrary. Each party receives as an input the labels of the sender and receiver, here $(s=7,r=3)$. The sender $s$ has the additional input $x$, and the receiver is the only party who produces an output ($a$). Different causal constraints limit the set of attainable correlations $P_{A|S,R,X}$ in different ways.
• Figure 4: (a) Example of a graphical game $\bm w$ as a $2$-cycle on three vertices. Here, and in the other digraphs, we omit the vertex labels. (b) A target digraph $\bm r$ with weights. (c) The "inner product" amounts to the value $1$.
• Figure 5: Two classes of digraphs that describe graphical games for $n=15$ parties: (a) the $6$-fence digraph of order $15$, and (b) the $5$-Möbius digraph of order $15$.

Theorems & Definitions (36)

• Definition 1: Single-output scenario
• Definition 2: Operational 0/1 polytope, $\Delta_n$
• Lemma 1: Lifting tselentis2023
• proof
• Lemma 2: Facet generation
• proof
• Definition 3: Graphical game, graphical test
• Definition 4: Static causal order, $\mathcal{C}_n^\text{static}$
• Theorem 1: DAGs tselentis2023
• Definition 5: DAG polytope, $\mathcal{Q}_n^\text{DAG}$