A Brunnian Theorem for Finite Families of Random Variables

TL;DR

This paper proves that any finite connectivity structure can be represented by a family of random variables, extending the understanding of dependency relations in quantum entanglement studies.

Contribution

It provides the first rigorous proof that every finite connectivity structure corresponds to a family of random variables, confirming a conjecture from prior intuitive discussions.

Findings

Every finite connectivity structure can be realized by a family of random variables.

The proof solidifies the link between connectivity structures and dependency relations.

Supports the study of quantum entanglement by formalizing the structure of variable dependencies.

Abstract

In 2014, during a study on the connectivity structures of quantum entanglement, I specifically introduced the notion of ''the connectivity structure of a family of random variables'' -- a structure that expresses the dependency relations between the variables in question -- and I stated the following proposition, which can be described as Brunnian in reference to Hermann Brunn's work on links (1892) : "Every finite connectivity structure is that of a family of random variables". At the time, however, I neglected to write down the proof of this assertion, merely providing an intuitive idea of it. The purpose of this article is to present such a proof.