Reconstructing quantum theory from its possibilistic operational formalism

Eric Buffenoir (1)

1 INPHYNI - Institut de Physique de Nice

We develop a possibilistic semantic formalism for quantum phenomena from an operational perspective. This semantic system is based on a Chu duality between preparation processes and yes/no tests, the target space being a three-valued set equipped with an informational interpretation. \\ A basic set of axioms is introduced for the space of states; it suffices to constrain this space to be a locally boolean qualitative domain. The subset of pure states is then characterized within this domain structure.\\ After having specified the notions of properties and measurements, we explore the notion of compatibility between measurements and of minimally disturbing measurements. The general existence of such measurements is then emphasized as the last key axiom to achieve the characterization of the domain structure on the space of states. This domain structure appears to be a new interesting generalization of 'concrete domains'.\\ An orthogonality relation is then defined on the space of states and its properties are studied. Equipped with this relation, the ortho-poset of ortho-closed subsets of pure states inherits naturally a structure of Hilbert lattice.\\ Finally, the symmetries of the system are characterized as a general subclass of Chu morphisms. We prove that these Chu symmetries preserve the class of minimally disturbing measurements and the orthogonality relation between states. These symmetries lead naturally to the ortho-morphisms of Hilbert lattice defined on the set of ortho-closed subsets of pure states.

Mots clés: Logical foundations of quantum mechanics; Preorders; orders; domains and lattices (viewed as categories); Categorical semantics of formal languages

Domaines: Physique Quantique [quant-ph]

Eric Buffenoir. Reconstructing quantum theory from its possibilistic operational formalism. 2021. ⟨hal-03052811v3⟩