Multipartite Causal Correlations: Polytopes and Inequalities

TL;DR

This paper characterizes the set of multipartite causal correlations as convex polytopes, identifies their facet inequalities, and demonstrates violations within quantum process matrix formalism, advancing understanding of causal structures in quantum physics.

Contribution

It provides a complete characterization of tripartite causal correlation polytopes and their inequalities, extending to more parties and linking to quantum violations.

Findings

Tripartite correlation set forms a convex polytope with deterministic vertices.

Facet inequalities define causal constraints and can be violated quantum mechanically.

Generalizations of inequalities apply to scenarios with more parties.

Abstract

We consider the most general correlations that can be obtained by a group of parties whose causal relations are well-defined, although possibly probabilistic and dependent on past parties' operations. We show that, for any fixed number of parties and inputs and outputs for each party, the set of such correlations forms a convex polytope, whose vertices correspond to deterministic strategies, and whose (nontrivial) facets define so-called causal inequalities. We completely characterize the simplest tripartite polytope in terms of its facet inequalities, propose generalizations of some inequalities to scenarios with more parties, and show that our tripartite inequalities can be violated within the process matrix formalism, where quantum mechanics is locally valid but no global causal structure is assumed.