Logical Methods in Computer Science Volume 15, Issue 2, 2019, pp. 3:1–3:51 Submitted Apr. 25, 2017 https://lmcs.episciences.org/ Published Apr. 26, 2019 GENERALISED MERMIN-TYPE NON-LOCALITY ARGUMENTS STEFANO GOGIOSO AND WILLIAM ZENG Quantum Group, University of Oxford, UK e-mail address:
[email protected] Rigetti Computing, Berkeley, CA e-mail address:
[email protected] 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, Buˇzekand 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. Introduction Non-locality is a defining feature of quantum mechanics, and its connection to the structure of phase groups is a key foundational question. A particularly crisp example of this connection is given by Mermin's argument for qubit GHZ states [Mer90], which finds practical application in the HBB quantum secret sharing protocol. In Mermin's argument, N qubits are prepared in a GHZ state (with Pauli Z as compu- tational basis), then a controlled phase gate is applied to each, followed by measurement in the Pauli X observable. Even though the N outcomes (each valued in Z2) are probabilistic, their parity turns out to satisfy certain deterministic equations.