From Operational Chu Duality to Coalgebraic Quantum Symmetry

Abstract

We pursue the principles of duality and symmetry building upon Pratt’s idea of the Stone Gamut and Abramsky’s representations of quantum systems. In the first part of the paper, we first observe that the Chu space representation of quantum systems leads us to an operational form of state-observable duality, and then show via the Chu space formalism enriched with a generic concept of closure conditions that such operational dualities (which we call “T₁-type” as opposed to “sober-type”) actually exist in fairly diverse contexts (topology, measurable spaces, and domain theory, to name but a few). The universal form of T₁-type dualities between point-set and point-free spaces is described in terms of Chu spaces and closure conditions. From the duality-theoretical perspective, in the second part, we improve upon Abramsky’s “fibred” coalgebraic representation of quantum symmetries, thereby obtaining a finer, “purely” coalgebraic representation: our representing category is properly smaller than Abramsky’s, but still large enough to accommodate the quantum symmetry groupoid. Among several features, our representation reduces Abramsky’s two-step construction of his representing category into a simpler one-step one, thus rendering the Grothendieck construction redundant. Our purely coalgebraic representation stems from replacing the category of sets in Abramsky’s representation with the category of closure spaces in the light of the state-observable duality telling us that closure is a right perspective on quantum state spaces.