Graphical Tests of Causality

TL;DR

This paper introduces simple graphical inequalities that serve as bounds on the success probability in communication games under various causal order constraints, extending Bell inequalities to multiple parties.

Contribution

It derives new inequalities for multiple parties under different causal constraints and defines weakly causal correlations, with a polynomial-time decision procedure.

Findings

Derived inequalities for static, definite, and bi-causal orders.

Defined weakly causal correlations satisfying kefalopoda inequalities.

Polynomial-time algorithm for testing weak causality.

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.