Mermin Non-Locality in Abstract Process Theories

ABSTRACT

The study of non-locality is fundamental to the understanding of quantum mechanics. The past 50 years have seen a number of non-locality proofs, but its fundamental building blocks, and the exact role it plays in quantum protocols, have remained elusive. In this paper, we focus on a particular flavour of non-locality, generalising Mermin's argument on the GHZ state. Using strongly complementary (aka canonically commuting) observables, we provide necessary and sufficient conditions for Mermin non-locality in abstract process theories. We show that the existence of more phases than classical points (aka eigenstates) is not sufficient, and that the key to Mermin non-locality lies in the presence of certain algebraically non-trivial phases. This allows us to show that the category fRel of finite sets and relations, a favourite toy model for categorical quantum mechanics, is Mermin-local. By considering the role it plays in the security of the HBB CQ (N,N) family of Quantum Secret Sharing protocols, we argue that Mermin non-locality should be seen as a resource in quantum protocols. Finally, we challenge the unspoken assumption that the measurements involved in Mermin-type scenarios should be complementary, opening the doors to a much wider class of potential experimental setups than currently employed.