Outer approximations of classical multi-network correlations

TL;DR

This paper introduces the postselected inflation framework, a method for approximating the set of classical multi-network correlations, providing tools for certification and extending existing inflation hierarchy results.

Contribution

It presents a new framework that converges to the true set of correlations in multi-network scenarios and links it to the standard inflation framework, enhancing understanding and applicability.

Findings

Framework provides converging outer approximations.

Mathematically equivalent to the standard inflation framework.

Extends inflation hierarchy to multi-network scenarios.

Abstract

We propose a framework, named the postselected inflation framework, to obtain converging outer approximations of the sets of probability distributions that are compatible with classical multi-network scenarios. Here, a network is a bilayer directed acyclic graph with a layer of sources of classical randomness, a layer of agents, and edges specifying the connectivity between the agents and the sources. A multi-network scenario is a list of such networks, together with a specification of subsets of agents using the same strategy. An outer approximation of the set of multi-network correlations provides means to certify the infeasibility of a list of agent outcome distributions. We furthermore show that the postselected inflation framework is mathematically equivalent to the standard inflation framework: in that respect, our results allow to gain further insights into the convergence proof of the inflation hierarchy of Navascu\`es and Wolfe [arXiv:1707.06476], and extend it to the case of multi-network scenarios.