Universal bound on the cardinality of local hidden variables in networks

TL;DR

This paper establishes a finite upper bound on the size of local hidden variables in complex networks, showing that local correlation sets are well-structured and describable by polynomial inequalities.

Contribution

It introduces an algebraic framework for local correlations in networks with multiple sources and proves a finite bound on hidden variable cardinality, extending Bell scenario analysis.

Findings

Sets of local correlations are connected and closed.

Local correlation sets are semialgebraic.

Derived tight polynomial Bell-like inequalities.

Abstract

We present an algebraic description of the sets of local correlations in arbitrary networks, when the parties have finite inputs and outputs. We consider networks generalizing the usual Bell scenarios by the presence of multiple uncorrelated sources. We prove a finite upper bound on the cardinality of the value sets of the local hidden variables. Consequently, we find that the sets of local correlations are connected, closed and semialgebraic, and bounded by tight polynomial Bell-like inequalities.