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joint work with Samson Abramsky (UCL).
Partial Boolean algebras (pBAs) were introduced by Kochen and Specker in their seminal work on contextuality, a key signature of non-classicality in quantum mechanics which has more recently been linked to quantum computational advantage. They provide a natural (algebraic-)logical setting for contextual systems, an alternative to traditional quantum logic à la Birkhoff-von Neumann in which operations such as conjunction and disjunction are partial, being only defined in the domain where they are physically meaningful (e.g. only for commuting projectors on a Hilbert space).
As a first step in the study of Stone-type dualities in this partial-algebraic setting, we extend the classical Tarski duality between sets and complete atomic Boolean algebras (CABAs). We establish a dual equivalence between the category of transitive partial CABAs and a category of exclusivity graphs. Such graphs are interpreted as spaces of possible worlds of maximal information, with edges representing logical exclusivity. The classical case appears as the complete graphs, since all possible worlds are mutually exclusive. Correspondingly, morphisms are relaxed from functions (between sets) to certain kinds of relations (between exclusivity graphs). The result implies that any transitive partial CABA can be recovered from its graph of atoms as an algebra of cliques (i.e. sets of pairwise exclusive worlds) modulo the identification of cliques that jointly exclude the same set of worlds.
This 'quantum' extension of the simplest Stone-type duality reveals a connection between Kochen and Specker's algebraic-logical setting of partial Boolean algebra and modern approaches to contextuality. Under it, a transitive partial CABA witnessing contextuality, in the Kochen-Specker sense of having no homomorphism to the two-element Boolean algebra, corresponds to an exclusivity graph with no 'points', i.e. with no maps from the singleton graph, equivalently described as stable, maximum-clique transversal sets.
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