A missing causal principle: Coordination

TL;DR

This paper introduces the coordination principle, a causal rule for perfect coordination among parties, and demonstrates its validity in quantum theory through Bell-like inequalities and a quantum coordination task.

Contribution

It establishes a multipartite causal principle, proves its satisfaction in quantum information, and introduces a quantum coordination task involving GHZ states.

Findings

Quantum theory satisfies the coordination principle in any network.

Bell-like inequalities certify the presence of a quantum common cause.

A quantum coordination task with GHZ states requires a quantum common cause.

Abstract

We introduce the coordination principle, which states that perfect coordination, in the form of agreement on a uniformly random output, among N parties is possible only if they share a common cause. This principle is purely causal and can be viewed as a multipartite generalization of Reichenbach's common cause principle. We prove that quantum information theory satisfy the coordination principle in any network, and derive noise-tolerant Bell-like inequalities that certify the presence of a common cause. We further show that the principle is not a consequence of no-signaling and independence alone by constructing a concrete operational probabilistic theory that obeys both principles while still allowing perfect coordination without a common cause. This possibility arises only in fully general causal scenarios with intermediate transformations between preparations and measurements. We also formulate a genuinely quantum coordination task, showing that the preparation of a multipartite GHZ state requires a quantum common cause, which can be certified by Bell-like inequalities which are experimentally testable. Finally, we discuss the open problem of finding a quantitative, noise-tolerant version of the coordination principle that constrains approximate coordination in any reasonable causal theory. This work is the extended version of the more compact letter and provides all the technical details of the proofs.