On the connection between the categorical and the modal logic approaches to Quantum Mechanics

Olímpia Lombardi, Matías Daniel Pasqualini, Juan Pablo Jorge +1 more

John Harding

Herman Dishkant

Peter Mittelstaedt

Giovanni Cinà

The study of the foundations of Quantum Mechanics, especially after the advent of Quantum Computation and Information, has benefited from the application of category-theoretic tools and modal logics to the analysis of Quantum processes: we witness a wealth of theoretical frameworks casted in either of the two languages. This paper explores the interplay of the two formalisms in the peculiar context of Quantum Theory. After a review of some influential abstract frameworks, we show how different modal logic frames can be extracted from the category of finite dimensional Hilbert spaces, connecting the Categorical Quantum Mechanics approach to some modal logics that have been proposed for Quantum Computing. We then apply a general version of the same technique to two other categorical frameworks, the ‘topos approach’ of Doering and Isham and the sheaf-theoretic work on contextuality by Abramsky and Brandenburger, suggesting how some key features can be expressed with modal languages.

Giovanni Cinà

This thesis aims at connecting the two research programs known as Categorical Quantum Mechanics and Dynamic Quantum Logic. This is achieved in three steps. First we define a procedure to extract a Modal Logic frame from a small category and a functor into the category of sets and relations. Second, we extend such methodology to locally small categories. Third, we apply it to the category of finite-dimensional Hilbert spaces to recover the semantics of Dynamic Quantum Logic. This process prompts new lines of research. At a general level, we study some logics arising from wide classes of small categories. In the case of Hilbert spaces, we investigate how to obtain richer semantics, containing probabilistic information. We design a logic for this semantics and prove that, via translation, it preserves the validities of Dynamic Quantum Logic.