Title: Dagger linear logic and Categorical Quantum Mechanics
In this talk, I will present an overview of my PhD research on the foundations of Categorical quantum mechanics. Categorical quantum mechanics uses the graphical calculus of compact closed categories to provide a rigorous and diagrammatic language for describing and reasoning about quantum processes between finite dimensional systems. Compact closed categories provide a categorical semantics for compact multiplicative linear logic with negation. Compact refers to the idea that the multiplicative conjunction and disjunction coincide, and that the truth and the false connective coincide in the fragment of the linear logic. In our research, we have developed a diagrammatic framework for CQM using linearly distributive and *-autonomous categories, with the aim of addressing the dimensionality constraint. *-autonomous categories provide the categorical semantics for (non-compact) multiplicative linear logic with negation. We proved that one can always recover the framework for finite dimensional settings from our more general framework. We generalize the existing algebraic structures of CQM to our new framework and study its implications. Surprisingly, we arrived at a connection between complementary systems of quantum mechanics and exponential modalities (operators which allow non-linear resources) of linear logic in our new general framework.