Generalised Mermin-type non-locality arguments

TL;DR

This paper generalizes Mermin-type non-locality arguments using group theory, providing new conditions for non-locality and extending quantum secret sharing schemes within a broad theoretical framework.

Contribution

It introduces a generalized framework for Mermin-type non-locality arguments and establishes algebraic conditions for non-locality, applicable beyond quantum theory.

Findings

New hierarchy of All-vs-Nothing non-locality arguments

Exact group-theoretic conditions for non-locality

Extension of quantum secret sharing protocols

Abstract

We broadly generalise Mermin-type arguments on GHZ states, and we provide exact group-theoretic conditions for non-locality to be achieved. Our results are of interest in quantum foundations, where they yield a new hierarchy of quantum-realisable All-vs-Nothing arguments. They are also of interest to quantum protocols, where they find immediate application to a non-trivial extension of the hybrid quantum-classical secret sharing scheme of Hillery, Bu\v{z}ek and Berthiaume (HBB). Our proofs are carried out in the graphical language of string diagrams for dagger compact categories, and their validity extends beyond quantum theory to any theory featuring the relevant algebraic structures.