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.
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.
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.
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.
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.
Contextuality is a key resource in quantum information and the device-independent security of quantum algorithms. In this work, we show that the recently developed, operational Mermin non-locality arguments provide a large, novel family of quantum realisable All-vs-Nothing models. In particular, they result in a diverse wealth of quantum realisable models which are maximally contextual (i.e. lie on the faces of the no-signalling polytope with no local elements), and could be used as a resource for the security of a new class of quantum secret sharing algorithms.
Categorical quantum mechanics studies quantum theory in the framework of dagger-compact closed categories. Using this framework, we establish a tight relationship between two key quantum theoretical notions: non-locality and complementarity. In particular, we establish a direct connection between Mermin-type non-locality scenarios, which we generalise to an arbitrary number of parties, using systems of arbitrary dimension, and performing arbitrary measurements, and a new stronger notion of complementarity which we introduce here. Our derivation of the fact that strong complementarity is a necessary condition for a Mermin scenario provides a crisp operational interpretation for strong complementarity. We also provide a complete classification of strongly complementary observables for quantum theory, something which has not yet been achieved for ordinary complementarity. Since our main results are expressed in the (diagrammatic) language of dagger-compact categories, they can be applied outside of quantum theory, in any setting which supports the purely algebraic notion of strongly complementary observables. We have therefore introduced a method for discussing non-locality in a wide variety of models in addition to quantum theory. The diagrammatic calculus substantially simplifies (and sometimes even trivialises) many of the derivations, and provides new insights. In particular, the diagrammatic computation of correlations clearly shows how local measurements interact to yield a global overall effect. In other words, we depict non-locality.
The participant attack is the most serious threat for quantum secret-sharing protocols. We present a method to analyze the security of quantum secret-sharing protocols against this kind of attack taking the scheme of Hillery, Bu\ifmmode \check{z}\else \v{z}\fi{}ek, and Berthiaume (HBB) [Phys. Rev. A 59, 1829 (1999)] as an example. By distinguishing between two mixed states, we derive the necessary and sufficient conditions under which a dishonest participant can attain all the information without introducing any error, which shows that the HBB protocol is insecure against dishonest participants. It is easy to verify that the attack scheme of Karlsson, Koashi, and Imoto [Phys. Rev. A 59, 162 (1999)] is a special example of our results. To demonstrate our results further, we construct an explicit attack scheme according to the necessary and sufficient conditions. Our work completes the security analysis of the HBB protocol, and the method presented may be useful for the analysis of other similar protocols.