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, has 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 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 fRel, a favourite toy model for categorical quantum mechanics, is Mermin local. We show Mermin non-locality to be the key resource ensuring the device-independent security of the HBB CQ (N,N) family of Quantum Secret Sharing protocols. Finally, we challenge the unspoken assumption that the measurements involved in Mermin-type scenarios should be complementary (like the pair X,Y ), opening the doors to a much wider class of potential experimental setups than currently employed. In short, we give conditions for Mermin non-locality tests on any number of systems, where each party has an arbitrary number of measurement choices, where each measurement has an arbitrary number of outcomes and further, that works in any abstract process theory.

Spectral Presheaves, Kochen-Specker Contextuality, and Quantale-Valued Relations

“…This non-contextuality result for Rel Q is consistent with a theorem which states the category of finite sets and relations is Mermin-local [14], lending some credibility to our definition of Kochen-Specker contextuality. We now show a partial converse of Theorem 4.1, that is, every global section of Spec P isolates some x ∈ X , although we do not claim that every global section is of the formx as defined in the proof of Theorem 4.1.…”

“…This is useful for exploring interdependencies of quantum or quantum-like phenomena, for example the many notions of non-locality and contextuality. In particular, in [14] an abstract notion of Mermin-locality is formulated in the language of Frobenius algebras, and the category of finite sets and relations FRel is shown to be Mermin-local.…”

Categorical Quantum Dynamics

“…A number of applications of CQM rely on strong complementarity as their active ingredient: most relevant are its appearance as an abstract version of the quantum Fourier transform in group-theoretic quantum algorithms [Vic12b,ZV14,Zen15,GZ15a], and the role it plays in connecting Mermin-type non-locality scenarios to the structure of phase groups in abstract process theories [CDKW12,CES10,GZ15b].…”

Abstract structure of unitary oracles for quantum algorithms

“…In Section 4.1.2 we cover the definition of strong complementarity [31], which has been used in the foundations of quantum mechanics to study non-locality [32,47] (Chapter 5), quantum secret sharing [47,114] (Section 5.4), and blackbox quantum algorithms [104,115,116] (Section 4.2). This allows a generalization beyond FHilb to strongly complementarity pairs of a quasi-Special †-Frobenius Algebra ( †-qSFA or †-qSCFA if commutative) and a †-SCFA.…”